On the Derivation Algebras of Parabolic Lie Algebras with Applications to Zero Product Determined Algebras by Daniel Brice A dissertation submitted to the Graduate Faculty of Auburn University in partial fulfillment of the requirements for the Degree of Doctor of Philosophy Auburn, Alabama August 2, 2014 Keywords: Reductive Lie algebra, parabolic subalgebra, derivation, zero product determined algebra, direct sum decomposition Copyright 2014 by Daniel Brice Approved by Huajun Huang, Chair, Associate Professor of Mathematics Randall R. Holmes, Professor of Mathematics Tin-Yau Tam, Lloyd and Sandra Nix Endowed Professor of Mathematics Abstract This dissertation builds upon and extends previous work completed by the author and his advisor in [5]. A Lie algebra g is said to be zero product determined if for each bilinear map j : g g !V that satisfies j(x, y) = 0 whenever [x, y] = 0 there is a linear map f : [g,g] !V such that j(x, y) = f [x, y] for all x, y 2g. A derivation D on a Lie algebra g is a linear map D : g !g satisfying D [x, y] = D(x), y + x, D(y) for all x, y2g. Derg denotes the space of all derivations on the Lie algebra g, which itself forms a Lie algebra. The study of derivations forms part of the classical theory of Lie algebras and is well understood, though some work has been done recently that generalizes some of the classical theory [9, 10, 14, 23, 24, 27, 30, 31, 34, 37]. In contrast, the theory of zero product determined algebras is new, motivated by applications to analysis, and supports a growing body of literature [1, 4, 5, 11, 33]. In this dissertation, we add to this body of knowledge, studying the two concepts of derivations and of zero product determined algebras individually and in relation to each other. This dissertation contains two main results. Let K denote an algebraically-closed, characteristic-zero field. Let q be a parabolic subalgebra of a reductive Lie algebra g over K or R. First we prove a direct sum decomposition of Derq. Derq decomposes as the direct sum of ideals Derq = L adq, where L consists of all linear maps on q that map into the center of g and map [q,q] to 0. Second, we apply the decomposition, along with results of [5] and [33], to prove that q and Derq are zero product determined in the case that g is a Lie algebra over K. We conclude by discussing several possible directions for future research and by ap- plying the main results to providing tabular data for parabolic subalgebras of reductive Lie algebras of types A5, G2, and F4. ii Acknowledgments I?d like to thank the Department of Mathematics and Statistics at Auburn University for supporting me in my studies and research for the past six years. My time at Auburn has been wonderful and formative, and the people here have been nothing but welcoming and friendly. Chief among these is my advisor, Dr. Huajun Huang. Dr. Huang, thank you for your patience with me when I often made the same mistake several times. Thank you for making the effort to contact me when I would disappear for weeks at a time, swallowed up in a miasma of depression. Thank you for having faith in me which I often lacked. I certainly would not have had the emotional will to continue if not for your guidance. I?d like to thank Dr. Holmes and Dr. Tam, whose work has influenced mine, who have given me helpful input throughout the process of writing this dissertation, and from whom I learned my craft. Dr. Holmes, I hope that I make you proud when I say that yours is the primary influence on my mathematical writing style. Dr. Tam, you have made this dissertation possible by giving me the necessary mathematical tools. Thank you both. I?d also like to thank my undergraduate professors: Dr. Elliott who introduced me to algebra, Dr. Grzegorczyk who introduced me to geometry, and Dr. Garcia who intro- duced me to mathematical reasoning and proof. It was at CSU - Channel Islands that, under their expert tutelage, the elusive cognitive lubricant called mathematical maturity first began to ferment inside me. It would be impossible to name everyone who has made this dissertation possible. I will close by thanking my family and my friends. I?ve had a great time these past six years, thanks to all of you. Now, anybody up for a round of Spades? iii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Background and Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1 Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Derivations and the adjoint map . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3 Real Lie algebras and complexification . . . . . . . . . . . . . . . . . . . . . . 19 2.4 Root space decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.5 Restricted root space decomposition . . . . . . . . . . . . . . . . . . . . . . . 25 2.6 Parabolic subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.7 Langland?s decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.8 The center of a parabolic subalgebra . . . . . . . . . . . . . . . . . . . . . . . 35 3 Derivations of Parabolic Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . 38 3.1 The algebraically-closed, characteristic-zero case . . . . . . . . . . . . . . . . 38 3.2 The real case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.3 Corollaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4 Zero Product Determined Derivation Algebras . . . . . . . . . . . . . . . . . . . 56 4.1 Zero product determined algebras . . . . . . . . . . . . . . . . . . . . . . . . 58 4.2 Parabolic subalgebras of reductive Lie algebras . . . . . . . . . . . . . . . . . 59 4.3 Derivations of parabolic subalgebras . . . . . . . . . . . . . . . . . . . . . . . 64 5 Examples and Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 iv 5.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 5.1.1 Type A5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 5.1.2 Type G2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 5.1.3 Type F4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 5.2 Directions for future research . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 v List of Figures 1.1 Decomposition of qS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Block matrix form of derivations in L . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1 A2 root system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.2 sl(C3) root space decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.3 Standard parabolic subalgebras of sl(C3) . . . . . . . . . . . . . . . . . . . . . . 29 2.4 F0 where D0 =fa2g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.5 The parabolic subalgebra q gl(C6) corresponding to D0 =fa1, a2, a4g. . . . . 34 2.6 The Levi factor decomposition of qS sl(C6) . . . . . . . . . . . . . . . . . . . . 35 2.7 A representative member of lZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.1 Embedding of HomK(V1 ?+ V2, V1) in gl(V1 ?+ V2) . . . . . . . . . . . . . . . . . 41 4.1 Tensor definition of zero product determined . . . . . . . . . . . . . . . . . . . . . 59 4.2 G2 root system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 5.1 Cartan matrix and transpose inverse for Type A5 . . . . . . . . . . . . . . . . . . 70 5.2 Cartan matrix and transpose inverse for Type G2 . . . . . . . . . . . . . . . . . . 70 5.3 Cartan matrix and transpose inverse for Type F4 . . . . . . . . . . . . . . . . . . 72 5.4 Logical relations among various types of derivation-like maps . . . . . . . . . . 76 vi List of Tables 2.1 sl(C3) root spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2 Simple roots of sl(C6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.3 Root spaces of sl(C6) relative to h . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 5.1 Partial multiplication table for sl(C6) in terms ofH . . . . . . . . . . . . . . . . 69 5.2 Partial multiplication table for sl(C6) in terms ofT . . . . . . . . . . . . . . . . 69 5.3 T in terms ofHand as matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 5.4 Parabolic subalgebras of type A5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 5.5 Parabolic subalgebras of type G2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 5.6 Parabolic subalgebras of type F4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 vii Chapter 1 Introduction Let K be an algebraically-closed field of characteristic-zero. Let g be a reductive Lie algebra overKorR. Let q be a parabolic subalgebra of g. The purpose of this dissertation is, primarily, twofold: first, to construct a direct sum decomposition of the derivation algebra Derq; second, to use this direct sum decomposition to extend work done in [5] and [33] and show that Derq is zero product determined. An algebraAwith multiplication is said to be zero product determined if for each bilinear map j : A A ! V that satisfies j(x, y) = 0 whenever x y = 0 there is a linear map f : A2 ! V such that j(x, y) = f x y for all x, y 2 A. This is a relatively new concept, first appearing in [4] and expanded upon in [5, 11, 33] and others. In the initial paper, published in 2009, Bre?ar, Gra?i?c, and S?nchez Ortega proved that the full matrix algebra over a commutative ring is zero product determined when is the usual matrix multiplication or when is the Jordan product x y = xy + yx, and also that the general linear algebra gl over a field is zero product determined when is the Lie bracket x y = [x, y] = xy yx [4]. Gra?i?c expanded the pool of Lie algebras known to be zero product determined to include the classical algebras Bl, Cl, and Dl over an arbitrary field of characteristic not 2 [11]. In 2011, Wang et al. proved that the parabolic subalgebras of a simple Lie algebras over K are zero product determined for K algebraically-closed and characteristic-zero (in particular, the simple Lie algebras over K are zero product determined) [33]. Our present research began as an attempt to extend this result to parabolic subalgebras of reductive Lie algebras over K and over R and to their derivation algebras. 1 A derivation D on a Lie algebra g is a linear map D : g !g satisfying D [x, y] = D(x), y + x, D(y) for all x, y 2g. Denote by Derg the space of all derivations on the Lie algebra g, which itself forms a Lie algebra. The study of derivations forms part of the classical theory of Lie algebras. The classical theory of derivations can perhaps be said to begin with the well-known result that if g is a semisimple Lie algebra over a field of characteristic not equal to two, then any derivation of g is an inner derivation [13, 26], so in particular g = Derg in case g is semisimple charF6= 2. In 1955, Jacobson proved that a Lie algebra g is nilpotent if it has a derivation f 2 Derg that is nonsingular [14]. Dixmier and Lister provided an example in 1957 show- ing that the converse was not possible [9]. They constructed a particular Lie algebra L, showed that L is nilpotent, characterized the derivation algebra Der L through explicit computation, and showed that every derivation of L had a non-trivial kernel. Of note is that Dixmier and Lister explicitly decompose Der L in this particular case and arrive at results similar to our results for general parabolic subalgebras of reductive Lie algebras. T?g?, in 1961, proved a partial converse to our result in the special case when q = g [27]. By 1972, Leger and Luks ? working over an arbitrary field of characteristic not equal to two ? were able to show that all derivations of a Borel subalgebra b of a semisimple Lie algebra g are inner derivations, analogous to the known result for the semisimple algebra g itself [19]. More generally, their result applies to the class of Lie algebras g that can be expressed as the semidirect product g = aog0 where the subalgebra g0 is nilpotent and the ideal a is abelian and acts diagonally on g0 [19]. This wider class of Lie algebras includes Borel subalgebras but does not include parabolic subalgebras. Working independently, Tolpygo extended the result of Leger and Luks to apply to any parabolic subalgebra q between b and g, but only in case the scalar field is the complex numbers C [29]. The balance of the work done in this era provides characterizations of derivations of special classes of Lie algebras [10, 24, 28]. 2 The recent direction that work on derivations has taken has been to relax the defini- tion of Lie algebra to include consideration of Lie algebras that draw scalars from com- mutative rings rather than from fields and to attempt to reproduce as much of the clas- sical theory as can be , characterizing derivations of specific classes of such Lie algebras [23, 30, 31]. Zhang in 2008 takes a different approach, defining a new class of solvable Lie algebras over C and characterizing their derivation algebras [37]. Other work has been in the direction of considering certain maps that are similar to but may fail to be derivations [6, 8, 34]. Unrelated to zero product determined algebras, except perhaps inspired by the concept, Wang et al. recently defined a product zero deriva- tion of a Lie algebra g as a linear map f : g ! g satisfying [f(x), y] + [x, f(y)] = 0 whenever [x, y] = 0 [34]. In the aforementioned paper, the authors go on to characterize the product zero derivations of parabolic subalgebras q of simple Lie algebras over an algebraically-closed, characteristic-zero field, ultimately showing all product zero deriva- tions of q to be sums of inner derivations and scalar multiplication maps [34]. In papers appearing in 2011 and 2012, Chen et al. consider nonlinear maps satisfying derivability and nonlinear Lie triple derivations respectively [6, 8]. The authors characterize all such maps on parabolic subalgebras of a semisimple Lie algebra over C as the sums of inner derivations and certain maps called quasi-derivations that may fail to be linear [6, 8]. This dissertation serves two purposes: to expand on the results characterizing deriva- tions of classes of Lie algebras and to apply these results to further the study of zero prod- uct determined algebras. We prove, among other results, the following two theorems. Theorem. Let q be a parabolic subalgebra of a reductive Lie algebra g over an algebraically-closed, characteristic-zero field or over R. The derivation Lie algebra Derq decomposes as the direct sum of ideals Derq = L adq where L consists of all linear transformation mapping q into its center and mapping [q,q] to 0. 3 Theorem. Let q be a parabolic subalgebra of a reductive Lie algebra g over an algebraically-closed, characteristic-zero field. q and Derq are zero product determined. We began this research with the goal of proving the latter theorem, as it is a natural extension of results found in [33]. As a tool to this end, we required an understanding of the structure of Derq, especially as it relates to the structure of q. Considering the vast body of literature on derivations, we assumed that results on the structure of Derq for such a q would be readily available; however, a review of the literature found only partial results, as we summarized above. Without the necessary tools to proceed, developing those tools quickly overtook our work on zero product determined algebras and became a central part of our research in and of itself. The results are complete and satisfying, and enable us to pursue our original goals. The method of proof of the former theorem relies on utilization of the grading on q afforded by the root system F. In order to motivate the methods employed, we offer the following example. The reader is encouraged to keep this example in mind during the general treatment in chapter 3. Example 1. We consider the parabolic subalgebra q of g = gl(C6) consisting of block- upper-triangular matrices in block sizes 3, 2, 1 (see figure 1.1). We write gl(C6) = gZ gS, where gZ = CI and gS = sl(C6). We decompose q similarly: q = gZ qS, where qS = q\gS. coroot contained in t coroot contained in c Figure 1.1: Decomposition of qS 4 gS has root space decomposition gS = h ?+ i6=j Cei,j where h consists of traceless diagonal 6 6 matrices. It is well known that the coroots hi = eii ei+1,i+1 form a basis of h. We further decompose h into t ?+c, where t = Spanfh1, h2, h4gand c = Spanfh3, h5g (see figure 1.1). It follows that t = h\[q,q] and that q has the vector space direct sum decomposition q = gZ ?+c ?+[q,q]. In light of this decomposition, a linear transformation that sends q to gZ and sends [q,q] to 0 has the block matrix form illustrated by figure 1.2. 0 @ gZ c [q,q] gZ 0 c 0 0 0 [q,q] 0 0 0 1 A Figure 1.2: Block matrix form of derivations in L The claims of the two theorems ? that Derq = L adq and that Derq is zero product determined ? may then be explicitly verified via computation in this special case. The proofs of the theorems in general will rely on carrying out the same decomposition of q and the accompanying computations in abstract. A brief outline of this dissertation: Chapter 2 provides the necessary background def- initions and tools needed to understand the results in the sequel. Except where noted in section 2.8, this chapter does not contain original research and may be skimmed or even skipped by an experienced Lie algebraist. Chapter 3 proves the former theorem and sev- eral ancillary results, as mentioned, primarily relying on the root space decomposition. Chapter 4 formally introduces the notion of a zero product determined algebra, summa- rizes some of the known results, and extends those results. Chapter 5 contains a short discussion on possible directions in which to generalize the results of this dissertation for future research. 5 Before we begin, we shall make note of some conventions of terminology and nota- tion. For the convenience of the reader, we shall use the term proposition for any result that is not original to this dissertation. The terms lemma, theorem, and corollary are used for results appearing for the first time in this dissertation. R and C will denote the field of real numbers and the field of complex numbers, respectively. If F is a field, we will say that F is complex-like to mean that F is algebraically-closed and of characteristic-zero. Typically, K will be used to denote complex-like fields and F will be used to denote more general fields. If V is a vector space, we denote the identity map on V by idV. If V1 and V2 are vector spaces over the same field F, we denote by HomF(V1, V2) the set of F-linear maps from V1 to V2, which is itself a vector spaces over F. If V1 and V2 are both subspaces of a vector space V, intersect trivially, and together span V, we write V = V1 ?+ V2. If V is a vectorspace, W a subspace, and f a linear map f : V ! V, we say f stabilizes W, or W is f -invariant, to mean f(W) W. We write fjW to denote the restriction of f to W; namely, the linear map W ! V defined by fjW(v) = f(v) for each v 2 W. If V is a vector space whose scalar field is ambiguous or non-standard in some way, we write V = VF to note that F is the scalar field. For example, the notation C3 will denote the set of 3-entry column vectors with entries in C viewed as a 3-dimensional complex vector space, and the notation C3R will denote the same set viewed as a 6-dimensional vector space with real number scalars. If M is a matrix, Mt denotes the transpose of M. If M is a matrix with complex entries, M denotes the conjugate transpose of M, ie M = M t. The notation Tr M denotes the trace of a matrix or linear transformation M (ie, the sum of the diagonal entries of M or a matrix representing M, respectively). In case Tr M = 0 we may say M is traceless for brevity. The notation ei,j is used to denote the matrix with 1 in the i-th row, j-th column entry and zeros elsewhere. The notation In (or simply I if n is understood) denotes the n n identity matrix with 1-s along the main diagonal and zeros elsewhere. 6 Chapter 2 Background and Setting This chapter reviews the basic facts of the classical theory of Lie algebras which are required for an understanding of the subsequent discussion. Proofs are included as space permits. Where a particular definition, theorem, or proof is not cited in the line of the text, the reader is referred to any of the standard texts on the subject, eg, [2, 13, 15, 17, 21, 25]. 2.1 Lie algebras Definition 2.1. A Lie algebra is a vector space g over a field F together with a binary operation [ , ] : g g !g satisfying: 1. [ , ] is F-bilinear, ie, 8x, y, z2g,8a2F, [x + ay, z] = [x, z]+ a[y, z] and [x, y + az] = [x, y]+ a[x, z]; 2. [ , ] is alternating, ie, 8x2g, [x, x] = 0; and 3. [ , ] satisfies the Jacobi identity, ie, 8x, y, z2g, x,[y, z] + y,[z, x] + z,[x, y] = 0. Proposition 2.1. For all x, y2g, [x, y] = [y, x]. Proof. Consider [x + y, x + y]. By condition 2, [x + y, x + y] = 0 and by conditions 1 and 2, [x + y, x + y] = [x, x] + [x, y] + [y, x] + [y, y] = [x, y] + [y, x], so 0 = [x, y] + [y, x], or rather [x, y] = [y, x]. 7 Example 2. Real three-space R3, together with the familiar cross product defined by 0 BB BB @ x1 y1 z1 1 CC CC A 0 BB BB @ x2 y2 z2 1 CC CC A = 0 BB BB @ y1z2 y2z1 x2z1 x1z2 x1y2 x2y1 1 CC CC A is a Lie algebra. It is straightforward to verify that satisfies the three conditions required of [ , ] in definition 2.1. Example 3. Let n be some positive integer. A space of n n matricesMmay be endowed with a bracket product by defining 8M, N 2M, [M, N] = MN NM. IfMis closed under taking linear combinations of bracket products of its members, then Mis a Lie algebra. Example 4. Denote by so(n) the space of all n n matrices with entries in R satisfying M = Mt (ie. M is skew-symmetric). so(n) is closed under taking linear combinations of matrix bracket products MN NM, so so(n) is a Lie algebra under [M, N] defined in example 3. Example 5. A matrix M with complex entries is called Hermitian if M = M . M is called skew-Hermitian is M = M . If M is Hermitian (res. skew-Hermitian), complex scalar multiple of M may fail to be Hermitian (res. skew-Hermitian). In fact, for Hermitian M, the scalar multiple iM is skew-Hermitian, and vice versa. Because of this, the vector space consisting of Hermitian (res. skew-Hermitian) matrices are vector spaces over R, despite Hermitian (res. skew-Hermitian) matrices admitting complex entries. Denote by su(n) the space of all n n skew-Hermitian traceless matrices. su(n) is closed under the bracket [M, N] defined in examples 3, and as such forms a Lie algebra 8 over R. The space of Hermitian matrices is not closed under the bracket. In fact, if M and N are Hermitian, [M, N] is skew-Hermitian. Example 6. Let V be a vector space over a fieldF. Denote by gl(V) the space of allF-linear maps from V to V. Denote by sl(V) the subspace of gl(V) consisting of traceless linear maps. Define the bracket product [ , ] by 8f , g2gl(V), [f , g] = f g g f . Then, gl(V) is a Lie algebra, and sl(V) is a Lie algebra under the bracket restricted to sl(V) sl(V). If the dimension of V is n, and if a basis for V is chosen, then gl(V) and sl(V) are concretely realized as the space of all n n matrices with entries in F and the space of all n n traceless matrices with entries inF, respectively, and the bracket defined here agrees with the bracket defined in example 3. Example 7. The vector space Cn n consisting of n n matrices with complex entries can be considered a Lie algebra over C, as gl(Cn) with dimension n2, or as a Lie algebra over R, as gl(CnR) with dimension 2n2. Definition 2.1 does not exclude the possibility of considering infinite-dimensional Lie algebras or Lie algebras over prime-characteristic fields, as is the case with ? for exam- ple ? [16] and [26], respectively. In addition, recent work in the study of Lie algebras has relaxed the definition to include the consideration of Lie algebras with scalars from a com- mutative ring rather than a field, such as in [2, 4, 11, 23, 30, 31]. This dissertation follows none of the aforementioned directions. Instead, all of the Lie algebras considered in this dissertation are finite-dimensional vector spaces, drawing scalars from a characteristic- zero field such as R or C. If g is a Lie algebra and if A, B are subsets of g (written A, B g) we use the notation [A, B] to denote the F-linear span of members of g of the form [a, b] where a 2 A and 9 b2B. Taking the F-linear span is essential here, as the setf[a, b]ja2 A, b2Bgoften fails to be a linear subspace of g. Definition 2.2. h is a subalgebra of g (written h g) means that h is a linear subspace of g and that [h,h] h. a is an ideal of g (written a g) means that a is a subalgebra of g and that [a,g] a. In light of proposition 2.1, [a,g] = [g,a], so the condition that [a,g] a is equivalent to the condition that [g,a] a. Example 8. Let g be any Lie algebra. g g trivially. Moreover, the bracket product of two ideals is an idea, so [g,g] is an ideal and is called the derived algebra of g. Example 9. Consider gl(R3). Notice that so(3) gl(R3), and sl(R3) = [gl(R3),gl(R3)] gl(R3). A subalgebra a g induces an equivalence relation on g by partitioning g into cosets x +a =fx + aja2agfor each x 2g. The set of all such cosets has a natural structure as a Lie algebra if, and only if, a is an ideal. Definition 2.3. Let g be a Lie algebra and let a g. The quotient algebra g/a is the Lie algebra consisting of the set g/a =fx +ajx2gg together with the bracket [x +a, y +a] = [x, y]+a. We omit the verification that the bracket on g/a is well-defined, while we note that the proof relies on the fact that a is an ideal. Definition 2.4. A Lie algebra g is abelian when [g,g] = 0. The name is not accidental or arbitrary. In fact, ifMis a space of commuting matri- ces, then for any M, N 2Mwe have [M, N] = MN NM = MN MN = 0, so that 10 the term abelian used here agrees with the familiar usage of the term from group theory when used to mean commutative. Definition 2.5. The center of g (written gZ) is the set of all z2g such that [z,g] = 0. The notation [x, S] is undefined in the case we consider, where S g and x 2g, but we now fix this notation to mean [fxg, S], ie, the linear span of the bracket products [x, s] for fixed x2g and for all s2S. Proposition 2.2. gZ is abelian and gZ g. Proof. We omit the verification that gZ is a linear subspace of g. We have left to show that [gZ,gZ] = 0 and that [gZ,g] gZ. Both follow from the observation that [gZ,g] = 0. Definition 2.6. A Lie algebra g is simple when g is non-abelian and the only ideals of g are 0 and g itself. Example 10. Let dim V 2. gl(V) is not simple because (gl(V))Z = FidV (where idV is the identity map on V). sl(V), however, is simple. It is somewhat non-trivial to prove this, though a proof may be found in any of the standard texts. The simple Lie algebras were completely classified and enumerated, in case F is complex-like, by Killing and Cartan as early as the 1890?s [13]. The classification of all simple Lie algebras over R appears in [17]. Definition 2.7. Let g1 and g2 be Lie algebras over F. An F-linear map f : g1 ! g2 is called a homomorphism (or endomorphism in case g1 = g2) if 8x, y2g1, f [x, y] = f(x), f(y) . If in addition f is injective (ie, one-to-one) and surjective (ie, onto), then f is called an isomorphism (or automorphism in case g1 = g2) and g1 and g2 are said to be isomorphic, written g1 = g2. 11 Example 11. Let V be a vector space, and suppose that T : V ! V is a change of basis, represented my an invertible matrix A in the sense that T(v) = Av. Let g gl(V). There is a change of basis T0 : g !g corresponding to T, represented by matrix conjugation in that T0(x) = A 1xA. Then, the map T0 is an automorphism of g, since T0 [x, y] = A 1(xy yx)A = A 1xyA A 1yxA = (A 1xA)(A 1yA) (A 1yA)(A 1xA) = T0(x), T0(y) . Example 12. R3, is isomorphic to so(3) by the isomorphism r : R3 !so(3) defined by r 0 BB BB @ x y z 1 CC CC A = 0 BB BB @ 0 z y z 0 x y x 0 1 CC CC A . Definition 2.8. Let f : g1 ! g2 be a homomorphism. The kernel of f (written Ker f ) is the set Ker f =fx2g1jf(x) = 0g, and the image of f (written Im f ) is the set Im f =ff(x)2g2jx2g1g. Proposition 2.3. Let f : g1 !g2 be a homomorphism. Then the kernel of f is an ideal of g1, and the image of f is a subalgebra of g2. 12 The underlying vector space of a Lie algebra g may be decomposed as a direct sum of subspaces. For example g = h ?+ k if h and k are subspaces of g, if h\k = 0, and if Span(h[k) = g, but an otherwise unqualified decomposition does not reflect any of the Lie algebra structure of g, meaning that the vector space decomposition is not necessarily compatible in any meaningful way the with bracket. We define two notions of direct sums of Lie algebras that are, to various degrees, compatible with the bracket. Definition 2.9. Let g = a ?+b. g is said to be the Lie algebra direct sum of a and b, written g = a b if both a and b are ideals of g. The requirement that a and b be ideals of g guarantees that the bracket acts diagonally on either summand. In other words, [a,b] = 0 since [a,b] a\b = 0. In this way, g is thought of as taking two distinct Lie algebras, a and b, and combining them together with the natural componentwise bracket rule [a1 + b1, a2 + b2] = [a1, a2]| {z } 2a +[b1, b2]| {z } 2b for a1, a2 2a, b1, b2 2b. Definition 2.10. Let g be a Lie algebra. g is called semisimple if g is the direct sum of simple ideals. g is called reductive if g = gZ gS for some semisimple ideal gS. Semisimple and reductive Lie algebras are natural generalization of simple Lie al- gebras, in the sense that much of the theory of simple Lie algebras can be extended to semisimple and reductive Lie algebras. We state without proof that the semisimple ideal gS of a reductive g is maximal and unique up to isomorphism and that the simple sum- mands of a semisimple g are unique up to isomorphism. Example 13. sl(Cn) is semisimple because it is a sum of one simple ideal. gl(Cn) = CIn sl(Cn) is neither simple nor semisimple but is reductive, with center consisting of scalar 13 matrices CIn and maximal semisimple ideal sl(Cn) (Recall In denotes the n n identity matrix). Definition 2.11. Let g = a ?+ b. g is said to be the Lie algebra semidirect sum of a and b, written g = aob, if a is an ideal of g and b is a subalgebra of g. The notation g = aob is chosen to remind us that a g. Since a is an ideal, [b,a] a. Explicitly, for each x 2 b, the map x : a ! a defined by x a = [x, a] is a linear transformation on a. Because of this, we say that b acts on a and that g is an extension of b by a. Then the bracket rule on g may be described in terms of the brackets on the individual summands and the action of b on a as [a1 + b1, a2 + b2] = [a1, a2]+ b1 a2 b2 a1| {z } 2a +[b1, b2]| {z } 2b . 2.2 Derivations and the adjoint map Proposition 2.4. Let g = aob. Define the map r : b !gl(a) by r(x) = x (ie, r(x)(a) = [x, a]) for x2b. The map r is a homomorphism of Lie algebras. Proof. We must show r [x, y] = r(x), r(y) for all x, y2b. Let a2a. r [x, y] (a) = [[x, y], a] = [x,[y, a]] [y,[x, a]] by definition 2.1 = r(x) r(y) r(y) r(x) (a) = [r(x), r(y)] (a). Since a was arbitrary, r [x, y] = r(x), r(y) for all x, y2b. The map r is said to be a representation of b, and a is said to be a b-module. 14 Definition 2.12. Let g be a Lie algebra. A representation of g is a homomorphism r : g ! gl(V) for some (finite-dimensional) vector space V. V, in this case, is said to be a g-module. When r is injective it is said to be a faithful representation. Representations are a tool by which an abstract Lie algebra g may be studied more concretely by considering Lie algebras consisting of linear transformations. If a basis for V is chosen, the linear transformations themselves are then represented by matrices, further simplifying the study of g. Proposition 2.5 (Ado?s Theorem). Let F be a characteristic-zero field. Let g be a (finite- dimensional) Lie algebra over F. Then, g admits a faithful representation. Explicitly, g is iso- morphic to a space of matrices with entries in F and bracket [M, N] = MN NM [2, Ch. I, ?7.3]. Example 14. The map r of example 12 is a faithful representation of R3, onto so(3). Since r is an isomorphism, it has an inverse r 1 : so(3) ! R3; however, r 1 is not a representation because it does not map into a subspace of some gl(V). Definition 2.13. Let g be a Lie algebra. A linear map D : g !g is called a derivation if 8x, y2g, D [x, y] = D(x), y + x, D(y) . The definition of derivation is motivated by the familiar product rule of differenti- ation. In fact, the differential operator ddx is a derivation in a suitable context. We will not spend time developing this idea, other than to mention it. The interested reader is referred to [3] for an algebraic treatment or to [12] or [36] for a geometric point of view. Definition 2.14. For a Lie algebra g, Derg denotes the set of all derivations on g. We note that if D1 and D2 are derivations, D1 D2 need not necessarily be a deriva- tion; however, [D1, D2] = D1 D2 D2 D1 is a derivation. In light of this, we have Derg gl(g). 15 Definition 2.15. Let g be a Lie algebra. For x2g the adjoint of x is the map ad x : g !g defined by ad x(y) = [x, y] for all y2g. The adjoint map is the map ad : g !gl(V). Proposition 2.6. ad : g ! gl(g) is a representation (in particular, a Lie algebra homomor- phism). Moreover, for each x2g, ad x is a derivation on g. The proposition follows from definition 2.1. In a sense, the definition of a Lie algebra g is intended to ensure that the action of multiplication by an element x (ie, the map ad x) is a derivation on g for all x 2g. It is for this reason that derivations take a primary role in the study of Lie algebras. Proof of proposition 2.6. We omit the proofs that ad and ad x are linear maps. Let x, y, z2g and consider ad x. We must to show that ad x([y, z]) = [ad x(y), z]+[y, ad x(z)]. ad x [y, z] = x,[y, z] = z,[x, y] y,[z, x] by condition 3 = [x, y], z + y,[x, z] by condition 2 = ad x(y), z + y, ad x(z) , so ad x is a derivation. Next, we must show ad[x, y] = [ad x, ad y]. ad[x, y](z) = [x, y], z = [y, z], x [z, x], y by condition 3 = x,[y, z] y,[x, z] by condition 2 = ad x ad y(z) ad y ad x(z) = [ad x, ad y](z) so ad[x, y] = [ad x, ad y], that is, ad is a homomorphism. We write adg = Im ad and note that adg Derg. Furthermore, notice Ker ad = gZ. 16 Example 15. Consider the adjoint representation of gl(Cn). The kernel isCIn and the image is isomorphic to gl(Cn)/CIn = sl(Cn). In the example above, the traceless matrices sl(Cn) act as derivations on the space of matrices gl(Cn). A natural question that we will return to often in this dissertation is whether there are other derivations on gl(Cn), and if so, how we may characterize them. Definition 2.16. Let g be a Lie algebra. An inner derivation of g is a member of adg. Any member of Derg not in adg is called an outer derivation. Proposition 2.7. An inner derivation maps g into [g,g] and stabilizes ideals. Proof. Let D be an inner derivation, so D = ad x for some x 2 g. Let y 2 g be arbitrary and notice D(y) = ad x(y) = [x, y] 2 [g,g], verifying the first assertion. Next, let a g. D(a) = ad x(a) = [x,a] a by the definition of ideal. In general, for a Lie algebra g we observe the chain of subspaces adg Derg gl(g) consisting of inner derivations, all derivations, and all linear maps respectively. The next theorem, a classical result in the theory of Lie algebras, completely characterizes the derivations of semisimple Lie algebras. Our work in chapter 3 of this dissertation can be understood as a generalization and extension of this classical result. Proposition 2.8. Let g be a semisimple Lie algebra over a field F of characteristic not equal to two. The only derivations of g are inner derivations [13, 26]. The proposition states that Derg = adg in case g is semisimple. A large portion of our work is to characterize outer derivations when q is a parabolic subalgebra (cf. section 2.6) of a reductive g. 17 Let g be semisimple and write as the sum of its simple ideals g = g1 ... gk. Each simple gi is of course semisimple, so Dergi = adgi = gi/(gi)Z = gi by proposition 2.8 and since each (gi)Z = 0. Applying proposition 2.8 to the semisimple g gives Derg = adg = g/gZ = g = g1 ... gk so that we have Der(g1 ... gk) = Der(g1) ... Der(gk) in case each gi is simple. In this fashion, the direct sum structure of g is carried over to the derivation algebra Derg when g is semisimple. This is not universally applicable to all direct sums of Lie algebras ? it is true for semisimple Lie algebras because of propositions 2.7 and 2.8. An arbitrary derivation of a general Lie algebra does not necessarily stabilize ideals, and direct sum decomposition is not necessarily preserved. There are two ideals, however, that are stabilized by every derivation, related in the following proposition. Proposition 2.9. Let D be a derivation on an arbitrary Lie algebra g. D stabilizes [g,g] and gZ. Proof. Let x, y2g. D([x, y]) = [D(x), y]+[x, D(y)]2[g,g], so D stabilizes [g,g] as desired. Next, let z 2gZ. We need to show D(z)2gZ. Let x 2g and consider D([z, x]). 0 = D([z, x]|{z} =0 ) = [D(z), x]+[z, D(x)]| {z } =0 = [D(z), x], so [D(z), x] = 0 for all x2g, meaning D(z)2gZ as desired. 18 2.3 Real Lie algebras and complexification Let i denote the imaginary unit. Let g be a Lie algebra over R. g is an R-vector space, but it is possible that vectors in g admit complex entries (cf. example 5). We would like to define ?g as the vector space g + ig, but we are concerned about possible notational collisions between i and entries of members of g. However, in light of Ado?s Theorem (proposition 2.5), g is isomorphic to a real Lie algebra consisting of matrices with real entries, and we think of g in this way as we proceed in order to avoid this issue. The vector space ?g = g ?+ ig =fx + iyjx, y2gg of dimension 2 dimg over R may be thought of as a C-vector space of dimension dimg. We may define a bracket on ?g by [x + iy, u + iv] = [x, u] [y, v]| {z } 2g + i([x, v]+[y, u])| {z } 2ig . We may verify that ?g, together with the bracket defined above, satisfies definition 2.1 with F = C, making ?g a Lie algebra over C. Definition 2.17. The complex Lie algebra ?g is called the complexification of the real Lie algebra g. g is called a real form of ?g. It is possible for non-isomorphic real Lie algebras g1 and g2 to have isomorphic com- plexifications bg1 = bg2 = g. In that case, both g1 and g2 are real forms of the complex Lie algebra g. Example 16. Since sl(Rn) ?+ isl(Rn) = sl(CnR) = su(n) ?+ isu(n), both sl(Rn) and su(n) are real forms of sl(Cn). Proposition 2.10. Let g be real, let ?g = g ?+ ig be the complexification of g. Then the center of ?g is the complexification of the center of g, namely ?gZ = cgZ = gZ ?+ igZ. 19 Proof. Let z 2 ?gZ. Write z = x + iy with x, y 2g. Now, for arbitrary w = u + iv 2 ?g with u, v2g we have 0 = [z, w] = [x + iy, u + iv] = [x, u] [y, v]+ i([x, v]+[y, u]) and by direct sum decomposition [x, u] = [y, v] and [x, v] = [y, u]. Adding these equa- tions gives 8u, v2g, [x, u + v] = [y, v u] (2.1) Setting v = u in equation 2.1 produces [x, 2u] = 0 for all u 2g, so x 2gZ. Similarly, setting u = v in equation 2.1 produces 0 = [y, 2v] for all v2g, so y2gZ. Proposition 2.11. Let g be a semisimple (res. reductive) Lie algebra over R. The complexification ?g of g is semisimple (res. reductive) [17, Ch. VI, ?9]. Proposition 2.12. Let D be a derivation of the real Lie algebra g. Then ?D defined by ?D(x + iy) = D(x)+ iD(y) is a derivation of ?g. Moreover, ?D stabilizes g. Proof. Let z = x + iy, w = u + iv be arbitrary elements of ?g. ?D([z, w]) = ?D [x + iy, u + iv] = ?D [x, u] [y, v]+ i [x, v]+[y, u] = D [x, u] [y, v] + iD [x, v]+[y, u] = D([x, u]) D([y, v])+ i D([x, v])+ D([y, u]) = [D(x), u]+[x, D(u)] [D(y), v] [y, D(v)] + i [D(x), v]+[x, D(v)]+[D(y), u]+[y, D(u)] 20 = [D(x), u + iv]+[x + iy, D(u)]+ i[x + iy, D(v)]+ i[D(y), u + iv] = [D(x), w]+ i[D(y), w]+[z, D(u)]+ i[z, D(v)] = [D(x)+ iD(y), w]+[z, D(u)+ iD(v)] = ?D(z), w + z, ?D(w) So ?D is a derivation on ?g. ?D stabilizes g by definition. Indeed, if x2g, then ?D(x) = ?D(x + i0) = D(x)2g. 2.4 Root space decomposition LetKdenote a complex-like field. Fix g as denoting a semisimple Lie algebra overK. We will show how to decompose g into a (vector space) direct sum of certain subspaces that have desirable interactions with the bracket [ , ], made explicit below. Our approach largely follows Humphreys, and the reader is referred to [13, Ch. II, ?8] for proofs. Definition 2.18. A toral subalgebra of g is a subalgebra h g that is abelian and for each x 2 h, the map ad x : g ! g is diagonalizable. If h is a maximal toral subalgebra, it is called a Cartan subalgebra of g. For a general Lie algebra g, a Cartan subalgebra is typically defined to be a self- normalizing nilpotent subalgebra, but this is more generality than we require. Within the class of semisimple Lie algebras, the general notion coincides with the simpler-to-state definition 2.18. Cartan subalgebras are unique in the sense made precise below. Proposition 2.13. The Cartan subalgebras of g are conjugate to one another, in the sense that any one may be transformed into any other by an appropriate automorphism of g. As a result of proposition 2.13, each Cartan subalgebra of g has the same dimension. We call this number the rank of g. Now, fix h as denoting a specific Cartan subalgebra of g. Consider the dual vector space h consisting of linear functionals a : h !C. 21 Definition 2.19. For a non-zero a2h , define the subspace ga of g by ga =fx2gj8h2h,[h, x] = a(h)xg. In case ga 6= 0, a is called a root and ga is called a root space. Denote by F the set of roots. F is called a root system. The rank of F is the dimension of h . Some basic facts about the root spaces ga for a2F: Proposition 2.14. Let a, b2F. [ga,gb] ga+b if a + b2F and [ga,gb] = 0 if a + b /2F. Moreover, each ga is one-dimensional. Let Z+ denote the set of positive integers and Z denote the set of negative integers. Follows are some basic facts about the geometric properties of the root system F. Proposition 2.15. Let r denote the rank of F. A set D of r roots (refered to as a base of F) may be selected so that each root b2F can be written uniquely as either a Z+-linear combination or a Z -linear combination of roots in D. Moreover, F = F and if b2F and b = ki=1 ai with each (not-necessarily non-repeating) ai 2D, we may rearrange the terms of the sum so the partial sums ji=1 as(i) lie in F for each j k each (where s denotes an appropriate permutation). In light of propositions 2.14 and 2.15 we fix the following notation and terminology: a D as in proposition 2.15 is called a base of F, and roots in D are called simple roots; roots in F generated by positive integer combinations of simple roots are called positive roots, and F+ denotes the set of positive roots; and similarly for negative roots and F . From each gb for b2F we arbitrarily choose an xb 2gb so that gb = Kxb by proposition 2.14. Proposition 2.16 (Root space decomposition). The semisimple Lie algebra g decomposes as the vector space direct sum g = h ?+ b2F Kxb. 22 This decomposition is called the root space decomposition of g relative of h, and is essentially unique in that the root space decompositions of g relative to two Cartan subalgebras h1 and h2 differ only by an automorphism of g. We may extend these notions to reductive Lie algebras over K. When g = gZ gS is reductive, then a Cartan subalgebra of g takes the form gZ h, where h is a Cartan subalgebra of gS. In this case, by root system F of the reductive g we mean the root system of the semisimple part gS, and by root space decomposition of g we mean the vector space direct sum decomposition g = gZ ?+h ?+ b2F Kxb. Example 17. Consider sl(C3), concretely the Lie algebra of 3 3 traceless matrices with complex entries. One choice of Cartan subalgebra is the subalgebra h consisting of di- agonal matrices. We take as basis for h the matrices h1 = e1,1 e2,2 and h2 = e2,2 e3,3. (Recall, ei,j denotes the matrix with a 1 in the i, j position and zeros elsewhere.) Our next task is to find all b 2 h such that gb 6= 0. Write b = b1h 1 + b2h 2, where h i (hj) = di,j is the dual functional to the vector hi. h 1 and h 2 are not roots, as gh 1 = gh 2 = 0. However, since h 1, h 2 together span h , any root b will be of the form b = b1h 1 + b2h 2 with b1, b2 2 C. In particular, straightforward computation show that g2h 1 h 2 6= 0 and g h 1+2h 2 6= 0, among others. We may write a1 = 2h 1 h 2, a2 = h 1 + 2h 2 and as we shall see below, D =fa1, a2gis a base of the root system F. We record the individual root spaces, our choice base D, and our choice of basis vector for each root space in table 2.1. Root b b in terms of D Root Space gb Choice of xb 2h 1 h 2 a1 Ce1,2 xa1 = e1,2 h 1 + 2h 2 a2 Ce2,3 xa2 = e2,3 h 1 + h 2 a1 +a2 Ce1,3 xa1+a2 = e1,3 2h 1 h 2 a1 Ce2,1 x a1 = e2,1 h 1 + 2h 2 a2 Ce3,2 x a2 = e3,2 h 1 + h 2 a1 a2 Ce3,1 x a1 a2 = e3,1 Table 2.1: sl(C3) root spaces 23 Because the root system F may be written F =fa1, a2, a1 +a2, a1, a2, a1 a2g, we see that D = fa1, a2g was a suitable choice of base. Since dimh = 2, F lies in a two-dimensional plane. Figure 2.1 provides an illustration of F. - A A A A AAU A A A A AAK a1 a2 a1 +a2 a1 a2 a1 a2 Figure 2.1: A2 root system Finally, sl(C3) decomposes as the vector space direct sum of root spaces: sl(C3) = h ?+ b2F Cxb. Refer to figure 2.2 for a graphical representation of the root space decomposition of sl(C3). 2 4 h ga1 ga1+a2 g a1 h ga2 g a1 a2 g a2 h 3 5 Figure 2.2: sl(C3) root space decomposition The root space decomposition of sl(C3) ? and more generally of sl(Cn) ? is, in a sense, a triviality, since it merely write sl(C3) in terms of the standard unit matrices ei,j. Partly this is because we chose a well-behaved Cartan subalgebra, but partly this 24 situation is intentional. The root space decomposition of sl(Cn) relative to the Cartan subalgebra consisting of diagonal matrices can be thought of as the motivating example for root space decomposition in general. In this sense, the root space decomposition of a general reductive g provides an sl-esque basis for general g [13], allowing one to reduce questions in Lie algebra theory to questions in linear algebra and matrix theory. Root space decomposition also allows for induction on the height of roots as a proof technique (where, for example, the roots a2 and a1 a2 have respective heights 1 and 2). 2.5 Restricted root space decomposition In this section, let g denote a semisimple Lie algebra over R. We will define the re- stricted root space decomposition of g, a coarser decomposition than the analogous root space decomposition for Lie algebras over complex-like fields. Except for some nota- tional changes made for internal consistency, the development of these ideas here follows Knapp?s, and the reader is referred to [17, Ch. VI, ?2-4] for proofs. Definition 2.20. Let q : g !g be an automorphism. q is called a Cartan involution of g if q q = idg and if the bilinear map Bq(x, y) = Tr ad x ad q(y) (2.2) is positive definite. For x, y 2 g, notice that ad x, ad q(y) are linear transformations g ! g, which we may think of as a matrices. Observing this, we see that the Bq, defined by equation 2.2, is well defined. The interested reader may note that, in general, the bilinear map B(x, y) = Tr(ad x)(ad y) on g g is called the Killing form, after Wilhelm Killing, and has many applications in the structure theory of Lie algebras [2, 13, 17]. We will not make use of the Killing form in the sequel, though Cartan involutions are necessary to arrive at the Cartan decomposition of g, described below. 25 Example 18. Recall sl(Rn), su(n), and sl(CnR) are Lie algebras over R. The map q(M) = Mt on sl(Rn) is a Cartan involution. The identity map on su(n) is a Cartan involution. Complex conjugation q(M) = M is a Cartan involution on sl(CnR), as well as is the map q(M) = M . Proposition 2.17. Every real Lie algebra g has a Cartan involution q. The Cartan involution q of g is essentially unique, in the sense that if q0 is another Cartan involution of g, then q0 = j q j 1 for some automorphism j of g. Given a Cartain involution q we may consider the eigenvalues of q. Since q q = idg the eigenvalues of q are 1 and 1. We write k for the eigenspace corresponding to 1 and p for the eigenspace corresponding to 1, and observe the following results: Proposition 2.18. For q a Cartan involution of g we have g = k ?+ p, with k and p as above. Furthermore, [k,k] k, [k,p] p, and [p,p] k. Definition 2.21. Notation as above, the decomposition g = k ?+ p is called the Cartan decomposition of g. Starting from a Cartan decomposition, we select a maximal abelian subspace a p and we set m =fx2kj[x,a] = 0g k. For each l2a we write gl =fx2gj8h2a,[h, x] = l(h)xg analogously to the complex-like case. Definition 2.22. Let l2a be non-zero. In case gl 6= 0, we call l a restricted root of g and we call gl the restricted root space associated to l. F = fl2a jl is a restricted rootg is called the restricted root system of g relative to a. Proposition 2.19 (Restricted root space decomposition). g decomposes as the vector space direct sum g = a ?+m ?+ l2F gl. 26 Furthermore, the restricted root spaces ga,gb satisfy [ga,gb] ga+b. In contrast to the complex-like case, the restricted root space gl need not be one- dimensional. The restricted root system exhibits some of the same geometric properties exhibited by the root system of a complex-like Lie algebra, which we discuss in the next two propositions. Proposition 2.20. The restricted root system F satisfies F = F. Furthermore, a set of positive restricted roots F+ may be selected with the properties that for each l 2F exactly one of l or l lies in F+ and for each a, b2F+, if a+ b2F then a+ b2F+. With F and F+ fixed, by a simple restricted root we mean a positive restricted root that does not decompose as the sum of two or more other positive restricted roots. Write D for the set of simple restricted roots. We call D a base of F. Proposition 2.21. There are dima simple restricted roots. D is linearly independent and spans F. F+ SpanZ+(D) and F SpanZ (D). If a, b2D, then a b /2F. In case g = gZ gS is reductive, then when we refer to the restrict root space de- composition or restricted root system of g, we mean respectively the restricted root space decomposition and restricted root system of gS. Example 19. Let g = sl(R3). With respect to the Cartan involution q(M) = Mt, the Car- tan decomposition g = k+p is given by k = so(3) and p = M2sl(R3) M is symmetric . a p may be chosen as consisting of diagonal traceless matrices, and then m = 0 since skew-symmetric matrices have zeros along the diagonal. For each pair (i, j) with i, j 3 and i6= j we have the restricted root li,j corresponding to the one-dimensional root space gli,j = Rei,j giving the restricted root space decomposition sl(R3) = a ?+ i,j 3 i6=j Rei,j similar to the root space decomposition of sl(C3) given in example 17. 27 Example 20. Consider g = sl(C3R) as a Lie algebra over R. The map q defined by q(M) = M is a Cartan involution on g with corresponding Cartan decomposition sl(C3R) = su(3) + isu(3) with k = su(3) and p = isu(3) (cf. example 5). Chose a p to consist of the diagonal matrices in p; namely, a consists of traceless diagonal matrices with pure imaginary entries. Then m consists of the diagonal matrices in k ? traceless diagonal matrices with real entries. For each pair (i, j) with i, j 3 and i 6= j we have a restricted root li,j where the corresponding restricted root space gli,j is 2-dimensional and takes the form gli,j = a1ei,j + a2iei,j a1, a2 2R . The restricted root space decomposition is then given by the vector space direct sum sl(C3R) = a ?+m ?+ i,j 3 i6=j gli,j. 2.6 Parabolic subalgebras In this section, we will define Borel subalgebras and parabolic subalgebras of re- ductive Lie algebras over a complex-like field K or over R. We first consider parabolic subalgebras in the case that g is semisimple over K and close by extending the definition to the larger class of Lie algebras. Definition 2.23. Let K be complex-like. Let g be semisimple over K. A Borel subalgebra b of g is a subalgebra of the form b = h ?+ b2F+ gb where h is a Cartan subalgebra of g and F is the root system of g relative to h. Example 21. In case g = sl(C3) (as a complex Lie algebra) and h consists of diagonal matrices, b is the subalgebra of g consisting of upper triangular matrices. 28 0 @ 1 A 0 @ 1 A 0 @ 1 A 0 @ 1 A q(?) = b q(fa1g) q(fa2g) q(fa1, a2g) = g Figure 2.3: Standard parabolic subalgebras of sl(C3) Definition 2.24. Let K be complex-like. Let g be semisimple over K. With notation as above, a standard parabolic subalgebra of g relative to the Cartan subalgebra h is a subalge- bra q of g satisfying b q g. A parabolic subalgebra of g is a subalgebra q g that is a standard parabolic subalgebra for some appropriate choice of h. Example 22. Where g = sl(C3) and b is as above, any standard parabolic subalgebras of g consists of block upper triangular matrices. Each parabolic subalgebra of g is simply one of the standard parabolic subalgebras conjugated by a change of basis of C3. Proposition 2.22. The standard parabolic subalgebras of g relative to b are in one-to-one cor- respondence with subsets of D. Explicitly, for a subset D0 D, the corresponding parabolic subalgebra q = q(D0) is q = h ?+ b2F0 gb where F0 = F+[ F\SpanD0 , ie, F0 consists of all positive roots and the negative roots spanned by D0. Example 23. Since D = fa1, a2g, g = sl(C3) has four standard parabolic subalgebras, illustrated in figure 2.3. Additionally, figure 2.4 gives an illustration of F0 when D0 = fa2g. That a particular root b is included in F0 is indicated by placing at the head of b, while b /2F0 is indicated with . 29 a1 - a1 a1 +a2 a 1 a2 A A A A AAU a 2 A A A A AAK a2 Figure 2.4: F0 where D0 =fa2g Since a parabolic subalgebra differs from a standard parabolic subalgebra only by an automorphism of g, proofs valid for any parabolic subalgebra q g need only con- sider the case where q is a standard parabolic subalgebra. We will often make use of this principle in the sequel. In case g = gZ gS is reductive over K, a Borel subalgebra b of g is of the form b = gZ bS where bS = b\gS is a Borel subalgebra of gS. A parabolic subalgebra q of g is of the form q = gZ qS where qS = q\gS is a parabolic subalgebra of gS. Having defined parabolic subalgbras in the complex-like semisimple and reductive cases, we now extend the definition to include real semisimple and reductive Lie algebras. Definition 2.25. Let g be a reductive Lie algebra over R. A parabolic subalgebra q of g is a subalgebra such that the complexification ?q is a parabolic subalgebra of ?g. Notice that because (?g)Z = d(gZ), a parabolic subalgebra q of a real reductive g = gZ gS has the form q = gZ qS, where qS is a parabolic subalgebra of g. In addition, parabolic subalgebras of a real Lie algebra exhibit a structure theory relating to the restricted root 30 space decomposition analogous to the structure theory of parabolic subalgebras of Lie algebras over a complex-like field. Proposition 2.23. Let g be reductive over R. Let q be a parabolic subalgebra of g. We may choose a restricted root space decomposition g = a ?+m ?+ l2F gl with restricted root system F and set of simple restricted roots D so that q has the form q = a ?+m ?+ l2F0 gl where F0 = F+[ F\SpanD0 , for an appropriate subset D0 of D. 2.7 Langland?s decomposition What follows is perhaps not part of the classical theory of Lie algebras, but can be found in chapter V, section 7 of [17] in case g is complex-like and in chapter VII, section 7 of [17] in case g is real. Let g be semisimple over a complex-like fieldKor overRand let q g be a parabolic subalgebra. Without loss of generality, we may assume that q arrises from a (restricted) root space decomposition of g. In the complex-like case, we have the following situation: g = h ?+ b2Fgb, where h is a Cartan subalgebra of g, F is the root system of g relative to h, 31 D is a base of F, D0 D is the subset of D corresponding to q, and F0 = F+[(F\SpanD0). Then q = h ?+ b2F0gb. Considering the case where g is real, we have the analogous situation: g = a ?+m ?+ l2Fgl where, F is the restricted root system of g relative to a, D is a set of simple restricted roots of F, D0 D is the subset of D corresponding to q, and F0 = F+[(F\SpanD0), so that q = a ?+m ?+ l2F0gl. F0 may be partitioned into two subsets, F0\ F and F0n F. This partition of F results in a vector space direct sum decomposition of q as q = l ?+n where l = h ?+ b2F0\ F gb and n = b2F0n F gb. Proposition 2.24. Notation as above, n is an ideal of q, l is a subalgebra of q, and l is reductive.[17, Ch. V, ?7; Ch. VII, ?7] 32 Definition 2.26. Notation as above, l is called the Levi factor of q and n is called the nilrad- ical of q. The decomposition q = l ?+n is called Langland?s decomposition. (Notice that since n is an ideal, q is the Lie algebra semidirect sum of the subalgebra l and the ideal n, and we may write q = lnn.) Finally, we extend this terminology and these notions to the case there g = gZ gS is reductive and q = gZ qS, by simply writing q = gZ (l ?+n) where l is the Levi factor and n is the nilpotent radical of qS. In such case, we say l (res. n) is the Levi factor (res. nilradical) of q and of qS interchangeably. Example 24. Let g = gl(C6) = CI6 sl(C6), let h consist of traceless diagonal matrices. Then h has dimension 5. The root system F will embed into a five-dimensional eucliedan space h , and so we need a base D consisting of five simple roots. Write hi = ei,i ei+1,i+1 for 1 i 5. Then fhig spans h and the dual functionals fh igspan h . By computing [hi, ej,j+1] for each pair (i, j)2f1, ..., 5g2 we find five simple roots D =fa1, ..., a5g, recorded in table 2.2. Root ai ai in terms offh ig Root space gai a1 (2, 1, 0, 0, 0) Ce1,2 a2 ( 1, 2, 1, 0, 0) Ce2,3 a3 (0, 1, 2, 1, 0) Ce3,4 a4 (0, 0, 1, 2, 1) Ce4,5 a5 (0, 0, 0, 1, 2) Ce5,6 Table 2.2: Simple roots of sl(C6) The root spaces of sl(C6) are listed in table 2.3. We enumerate each (positive) root b 2 F as a vector with respect to the basis D = fa1, ..., a5gand also with respect to the basisfh 1, ..., h 2g. 33 b in terms of D Root space gb Root space g b b in terms offh ig (1, 0, 0, 0, 0) Ce1,2 Ce2,1 (2, 1, 0, 0, 0) (0, 1, 0, 0, 0) Ce2,3 Ce3,2 ( 1, 2, 1, 0, 0) (0, 0, 1, 0, 0) Ce3,4 Ce4,3 (0, 1, 2, 1, 0) (0, 0, 0, 1, 0) Ce4,5 Ce5,4 (0, 0, 1, 2, 1) (0, 0, 0, 0, 1) Ce5,6 Ce6,5 (0, 0, 0, 1, 2) (1, 1, 0, 0, 0) Ce1,3 Ce3,1 (1, 1, 1, 0, 0) (0, 1, 1, 0, 0) Ce2,4 Ce4,2 ( 1, 1, 1, 1, 0) (0, 0, 1, 1, 0) Ce3,5 Ce5,3 (0, 1, 1, 1, 1) (0, 0, 0, 1, 1) Ce4,6 Ce6,4 (0, 0, 1, 1, 1) (1, 1, 1, 0, 0) Ce1,4 Ce4,1 (1, 0, 1, 1, 0) (0, 1, 1, 1, 0) Ce2,5 Ce5,2 ( 1, 1, 0, 1, 1) (0, 0, 1, 1, 1) Ce3,6 Ce6,3 (0, 1, 1, 0, 1) (1, 1, 1, 1, 0) Ce1,5 Ce1,5 (1, 0, 0, 1, 1) (0, 1, 1, 1, 1) Ce2,6 Ce2,6 ( 1, 1, 0, 0, 1) (1, 1, 1, 1, 1) Ce1,6 Ce6,1 (1, 0, 0, 0, 1) Table 2.3: Root spaces of sl(C6) relative to h Take D0 = fa1, a2, a4g. The standard parabolic subalgebra q corresponding to D0 consists of block upper triangular matrices corresponding to the partition 3 + 2 + 1 of 6, as illustrated in figure 2.5. q = 0 BB BB BB @ 1 CC CC CC A Figure 2.5: The parabolic subalgebra q gl(C6) corresponding to D0 =fa1, a2, a4g Consideration of the root systemFshows thatF\ F =f a1, a2, (a1 +a2), a4g. Fn F consists of the remaining positive roots. l (respectively n) consists of the block diagonal matrices (block strictly upper trian- gular matrices) in q that preserve the existing block structure, illustrated in figure 2.6. 34 l = 0 BB BB BB @ 1 CC CC CC A , n = 0 BB BB BB @ 1 CC CC CC A Figure 2.6: The Levi factor decomposition of qS sl(C6) As a reductive Lie algebra, l decomposes as l = lZ lS. Write hi = ei,i ei+1,i+1 for 1 i 5. Direct computation shows that center lZ is two dimensional: lZ =fa(h1 + 2h2 + 3h3 + 3h4)+ b(h4 + 2h5)ja, b2Cg. Perhaps more naturally, we may describe lZ in terms of matrices, illustrated in figure 2.7. l = 0 BB BB BB @ a a a b b 3a 2b 1 CC CC CC A Figure 2.7: A representative member of lZ A Cartan subalgebra of l is spanned by h1, h2, and h4, and lS is the Lie algebra direct sum of simple ideals isomorphic to sl(C3) and sl(C2) whereby l = C2 sl(C3) sl(C2). 2.8 The center of a parabolic subalgebra We conclude this chapter with a lemma ? characterizing the center of parabolic sub- algebras ? that will be required later. We did not find the following result in the standard texts, but it is elementary. As such, we presume it is already well-known, and we include it here rather than in chapter 3. 35 Lemma 2.25. Let q = gZ qS be a parabolic subalgebra of the reductive Lie algebra g = gZ gS over a complex-like field K or over R. The center of q is gZ. Proof. We consider first the special case where g = h ?+ b2F is semisimple over K. We assume without loss of generality that q is a standard parabolic subalgebra and write q = h ?+ b2F0gb. Let z2qZ so z = zh + b2F0 zgb. Then for any x2q we have 0 = [z, x] = [zh, x]+ b2F0 [zgb, x]. (2.3) Specifically, for 06= x2ga with a2F0, equation 2.3 becomes 0 = a(zh)x| {z } 2ga + b2F0 [zgb, x]| {z } 2ga+b and by direct sum decomposition 0 = a(zh) for all a 2F0. Since F0 spans h it must be the case that zh = 0, so z = b2F0 zgb. Now, let 06= h2h and apply equation 2.3. We see 0 = b2F0 [zgb, h] = b2F0 b(h)zgb which by direct sum decomposition yields 0 = b(h)zgb for all b2F0. Since h is arbitrary in h, zgb = 0 for all b, so z = 0. Having established that qZ = 0 when g is semisimple over K, that qZ = gZ when g is reductive over K follows from the Lie algebra direct sum decomposition q = gZ qS. We now consider the case where q is a parabolic subalgebra of a real reductive g. We have ?q is a parabolic subalgebra of ?g by definition. Then gZ + igZ = d(gZ) = (?g)Z = (?q)Z| {z } by above case = d(qZ) = qZ + iqZ. (2.4) 36 Finally, by Ado?s Theorem (proposition 2.5), we may assume that g consists of real matri- ces, so that we may seperate the real and imaginary part in equation 2.4, giving gZ = qZ, as desired. 37 Chapter 3 Derivations of Parabolic Lie Algebras We begin this chapter by stating the main result of this dissertation: Theorem. Let q be a parabolic subalgebra of a reductive Lie algebra g over R or over a complex- like field. Let L be the set of all linear transformations mapping q into qZ that send [q,q] to 0. Then L is an ideal of Derq and Derq decomposes as the direct sum of ideals Derq = L adq. Our main result is valid for R or for any complex-like field (such as C). However, the method of proof in the two cases is quiet different. The proof of the complex-like case is highly technical: given a derivation D on q, we explicitly construct a linear map L and an element x2q such that D = L +ad x, after which we prove that our construction satisfies the stated properties. The real case, in contrast, is high-level and abstract, appealing to the complex case of the central theorem as applied to ?q, the complexification of the real parabolic subalgebra q. Lie algebras over complex-like fields support a more regular structural decompo- sition then real Lie algebras affords. In particular, Langland?s decomposition ? while possible in the former case ? is completely unnecessary in order to prove theorem 3.1. Because of the less regular structure of real Lie algebras, Langland?s decomposition be- comes an important tool for the proof of theorem 3.4 3.1 The algebraically-closed, characteristic-zero case Throughout this section we use the following notational conventions: 38 K denotes a complex-like field; g = gZ gS denotes a reductive Lie algebra over K, where gZ is the center of g, and gS is the maximal semisimple ideal of g; q = gZ qS is a given parabolic subalgebra of g, where qS = q\gS is a parabolic subalgebra of gS. We choose a Cartan subalgebra h, a root system F, and a base D compatible with qS in the sense that qS is a standard parabolic subalgebra of gS relative to (h,F,D) and corresponds to a subset D0 D. Then qS = h ?+ a2F0 Kxa, where F0 = F+[ F\SpanD0 and where each xa is chosen arbitrarily from the one-dimensional root space it spans. Define t and c by t = h\[q,q] and c = Spanf[xa, x a]ja2DnD0g. Claim. h decomposes as h = c ?+t. Proof. Notice that h = Spanf[xa, x a]ja2Dg and that t = Spanf[xa, x a]ja2D0g. From these observations, we see that c\t = 0 and that Span(c[t) = h. 39 Noting that [q,q] = t ?+ a2F0Kxa, we arrive at the desired vector space direct-sum decompositions of q: q = gZ ?+h ?+ a2F0 Kxa = gZ ?+ hz}|{ c ?+t ?+ a2F0 Kxa | {z } [q,q] = gZ ?+c ?+[q,q]. We take a moment to note that alternatively c may have been chosen so that it coin- cides with the center of l in Langland?s decomposition q = l ?+n. This approach is not required in order to prove the complex-like case, but it is taken in order to simplify the proof of theorem 3.4 in the real case. For the remainder of the section, we assume all of the notational conventions de- scribed above without further mention, starting with a restatement of the central theorem in terms of the adopted notation. Theorem 3.1. For a parabolic subalgebra q = gZ qS of a reductive Lie algebra g = gZ gS over the complex-like field K, the derivation algebra Derq decomposes as the direct sum of ideals Derq = L adq, where L consists of all K-linear transformations on q mapping into qZ and mapping [q,q] to 0. Explicitly, for any root system F with respect to which q is a standard parabolic subalgebra, q decomposes as q = gZ ?+c ?+[q,q] and the ideal L consists of all K-linear transformations on q that map gZ +c into gZ and map [q,q] to 0, whereby Derq = Hom K (gZ ?+c,gZ) qS. 40 We must explain what we mean by HomK(gZ ?+c,gZ) as a Lie algebra, since it is merely a space of linear maps and does not come equipped with a Lie bracket by de- fault. For vector spaces V1, V2, we consider the space HomK(V2, V1) an abelien Lie alge- bra. Then, HomK(V1 ?+ V2, V1) may be realized as the Lie algebra semidirect sum Hom K (V1 ?+ V2, V1) = gl(V1)nHom K (V2, V1) with the action of gl(V1) on HomK(V2, V1) defined by f g = f g 8f 2gl(V1), g2Hom K (V2, V1). This definition is canonical in the sense that if we fix bases for V1 and V2, then HomK(V1 ?+ V2, V1) is identified with the subalgebra of gl(V1 ?+ V2) consisting of block matrices of the form illustrated in figure 3.1 (compare to figure 1.2), and the Lie bracket defined by the action above coincides with the standard Lie bracket of matrices, ie, [M, N] = MN NM. V1 V2V 1 V2 0 0 Figure 3.1: Embedding of HomK(V1 ?+ V2, V1) in gl(V1 ?+ V2) Proof of theorem 3.1. For clarity, the proof of the theorem is organized into a progression of claims. The first three claims establish that an arbitrary derivation may be written as a sum of an inner derivation and a derivation mapping gZ ?+c to gZ and [q,q] to 0. To this end, let D be an arbitrary derivation of q. Claim 1. There is an x 2 q such that D ad x maps c to gZ, annihilates t, and stabilizes each root space Kxa. Let h, k 2h be arbitrary and write D(h) = z + h0+ g ag(h)xg and D(k) = c + k0+ g ag(k)xg with z, c 2 gZ and h0, k0 2 h and ag(h), ag(k) 2 K. Recall [h, k] = 0 since 41 h, k2h and consider D([h, k]). 0 = D([h, k]) = [h, D(k)] [k, D(h)] = " h, c + k0+ g ag(k)xg # " k, z + h0+ g ag(h)xg # = " h, g ag(k)xg # " k, g ag(h)xg # = g ag(k)[h, xg] g ag(h)[k, xg] = g (ag(k)g(h) ag(h)g(k)) xg. So ag(k)g(h) ag(h)g(k) = 0 for all g2F0, h, k2h. (3.1) Furthermore, for any pair h, k for which g(h)6= 0 and g(k)6= 0, we have that ag(h) g(h) = ag(k) g(k) . This observation, along with the fact that g(h) 6= 0 for at least one h 2 h, allows us to associate with each g2F0 the numerical invariant dg = ag(h)g(h) independently of our choice of h. Notice that ag(h) dgg(h) = 0 by definition when g(h)6= 0. If g(h) = 0, the same equality still holds, as equation 3.1 becomes ag(h)g(k) = 0 for all k 2 h. Since at least one k 2 h satisfies g(k) 6= 0 we have ag(h) = 0 in case g(h) = 0, giving ag(h) dgg(h) = 0 for all h2h. (3.2) 42 Now, set x = g dgxg. Write D0 = D ad x. We will show that D0 maps c to gZ, annihilates t, and stabilizes each root space Kxa. We first show that D0 maps h to gZ ?+ h. Let h 2 h be arbitrary and again write D(h) = z + h0+ g ag(h)xg. We have that D0(h) = D(h) ad x(h) = z + h0+ g ag(h)xg g dg ad xg(h) = z + h0+ g ag(h)xg g dg ad h(xg) = z + h0+ g ag(h)xg g dgg(h)xg = z + h0+ g (ag(h) dgg(h))| {z } =0 by 3.2 xg = z + h0, affirming the assertion. Having established that D0 maps h into gZ ?+h, we have left to show that D0 annihi- lates t and stabilizes eachKxa. Let h2h and a2F0 be arbitrary, and write D0(h) = z + h0 and D0(xa) = c + k + g bgxg with z, c 2 gZ and h0, k 2 h and bg 2 K. Consider D0([h, xa]). On one hand, D0([h, xa]) = D0(a(h)xa) = a(h)D0(xa) = a(h)c +a(h)k + g a(h)bgxg. (]) 43 On the other hand, D0([h, xa]) = [D0(h), xa]+[h, D0(xa)] = [z + h0, xa]+ " h, c + k + g bgxg # = [h0, xa]+ g bg[h, xg] = a(h0)xa + g g(h)bgxg = a(h0)+a(h)ba xa + g6=a g(h)bgxg. ([) By equating ] and [ and by direct sum decomposition of q we obtain a(h)c = 0, (3.3) a(h)k = 0, (3.4) a(h)bg = g(h)bg for g6= a, and (3.5) a(h)ba = a(h0)+a(h)ba. (3.6) Since h is arbitrary, equations 3.3 and 3.4 give c = 0 and k = 0 respectively. Second, equations 3.5 give us bg(g a)(h) = 0 for all g 6= a. If any one bg 6= 0, then we would have g = a, a contradiction, so each bg = 0, whence D0 stabilizes each root space. Next, equation 3.6 gives us 0 = a(h0). Since a is arbitrary in F0 and F0 contains a basis of h , h0 = 0, so D0(h) gZ. Since derivations in general stabilize [q,q], D0(t) gZ\[q,q] = 0, so D0 annihilates t. The claim is verified. Claim 2. There is an h2h whereby D ad x ad h annihilates [q,q]. We have the D0 = D ad x maps c to gZ, annihilate t, and stabilize each root space Kxa. For each g2F0 write D0(xg) = cgxg 44 with cg 2K. Taking each a2D, the scalars ca define a linear functional ?c : h !C. We first verify that for each g2F0, cg = ?c(g). We begin with g2F0\F+. Let g = a1 + ... + ak with each ai 2D and where each sequential partial sum a1 + ... +ai 2F0. Then axg = [ [[xa1, xa2], xa3], , xak] for some 06= a2K. Apply D0 to both sides. Since D0 is a derivations, we have cgaxg = D0[ [[xa1, xa2], xa3], , xak] = D0(xa1), xa2 , xa3 , , xak + xa1, D0(xa2) , xa3 , , xak + [xa1, xa2], D0(xa3) , , xak + + [[xa1, xa2], xa3], , D0(xak) = ca1 [ [[xa1, xa2], xa3], , xak] + ca2 [ [[xa1, xa2], xa3], , xak] + ca3 [ [[xa1, xa2], xa3], , xak] + + cak [ [[xa1, xa2], xa3], , xak] = ca1 axg + ... + cak axg = ?c(a1 + ... +ak)axg = ?c(g)axg whereby cg = ?c(g) for all g2F0\F+. 45 Next let g2F0\F . Consider [xg, x g]2t, and apply D0. 0 = D0([xa, x a]) = [D0(xg), x g]+[xg, D0(x g)] = cg[xg, x g]+ c g[xg, x g] = (cg + c g)[xg, x g]. Since [xg, x g]6= 0, we have cg + c g = 0 so cg = c g = ?c( g) (since g2F0\F+) = ?c(g) as desired. Next, we use the canonical isomorphism Y : h ! h [22, Ch. VII, ?4] to produce Y(?c) = h2h. Notice that for each g2F0 we have the identity ?c(g) g(h) = 0 (3.7) by the definition of the canonical isomorphism. The claim is that D0 ad h annihilates [q,q]. Since [h,t] = 0, we need only check that D0 ad h maps each xg to 0. (D0 ad h)(xg) = ?c(g)xg g(h)xg = (?c(g) g(h))| {z } =0 by 3.7 xg = 0 46 verifying the claim. Claim 3. D = L + ad p for some p 2 q and some derivation L which maps gZ ?+c to gZ and maps [q,q] to 0. Set p = x + h as above and set L = D ad p = D0 ad h. Then D = L + ad p as desired. We note that since L is the difference of two derivations, L is itself a derivation. We know from claim 2 that L annihilates [q,q]. We must check that L maps gZ ?+c to gZ. We have already seen that gZ is the center of q, and more, that a derivation of q must stabilize the center of q. What is left to verify claim 3 is to check that L maps c into gZ. Let c2c be arbitrary. We have L(c) = D0 ad h (c) = D0(c) [h, c] = D0(c)|{z} 2gZ by claim 1 verifying claim 3. Since D was arbitrary, we now have that Derq is spanned by adq and the subset of Derq consisting of derivations that map gZ ?+c to gZ and [q,q] to 0. The next three claims establish facts about the relationship between these two sets. Claim 4. L Derq. L is defined as the set of K-linear endomorphisms of q mapping into the center of q and mapping [q,q] to 0. We will show that any such linear map is indeed a derivation of q. Suppose L : q !q is any K-linear map satisfying L(q) gZ and L([q,q]) = 0. Then, for any x, y2q we have [ 2gZz}|{ L(x), y]| {z } =0 +[x, 2gZz}|{ L(y)]| {z } =0 = 0 = L( 2[q,q]z}|{ [x, y])| {z } =0 47 so L is a derivation. Claim 5. L is an ideal of Derq. First we must show that L is closed under taking linear combinations of members. Let L1, L2 2L. We must show that L1 + kL2 2L (where k2K). Let x2q. We have (L1 + kL2)(x) = L1(x)|{z} 2gZ +k L2(x)|{z} 2gZ 2gZ so L1 + kL2 maps q into gZ. Next, let y2[q,q]. We have (L1 + kL2)(y) = L1(y)|{z} =0 +k L2(y)|{z} =0 = 0 so L1 + kL2 sends [q,q] to 0. Second, let L 2 L and D 2 Derq. We must show that [D, L] = D L L D 2 L, ie, that [D, L] maps q into gZ and maps [q,q] to 0. Recall that D(gZ) gZ and D([q,q]) [q,q]. Let x2q. Consider [D, L](x). [D, L](x) = D( 2gZz}|{ L(x))| {z } 2gZ L( 2qz}|{ D(x))| {z } 2gZ 2gZ so [D, L] maps q into gZ. Now, let y2[q,q] and consider [D, L](y). [D, L](y) = D( =0z}|{ L(y))| {z } =0 L( 2[q,q]z}|{ D(y))| {z } =0 = 0 so [D, L] maps [q,q] to 0?ie, [D, L]2L?verifying the claim. Claim 6. L and adq intersect trivially. 48 Suppose D 2 L\adq. Since D 2 L, D maps q into gZ. Since D 2 adq, D maps q into [q,q]. So, D maps q into gZ\[q,q] = 0, whereby D = 0, completing the proof of the theorem. As a simple application, we will use theorem 3.1 to derive a formula for the dimen- sion of Derq in terms of g and q. Corollary 3.2. For q a parabolic subalgebra of the reductive Lie algebra g = Cn gS over K, and with notation as above, the dimension of Derq is given by dim Derq = n +jDj jD0j n + dimqS. Proof. The corollary follows from the isomorphism Derq = HomK(gZ ?+c,gZ) adq, ie, the facts that for any vector spaces V1, V2 the dimension of V1 ?+ V2 is the sum dim V1 + dim V2 and the dimension of HomK(V1, V2) is the product (dim V1)(dim V2). Now, dimc = jDj jD0j, and dim adq = dimqS because adq = q/qZ = qS. Applying the mentioned general principles completes the proof. We will note now that a similar dimension-counting result will not be possible ? in general ? in the real case, though formulas may be possible for classes of certain parabolic subalgebras of specific real Lie algebras. Notice that the statement and proof of corollary 3.2 rely heavily on the explicit description of the ideal L and knowledge of the dimension of the subspace c q, which in turn rely on properties of the root space decomposition for Lie algebras over algebraically-close, characteristic-zero fields. Anal- ogous properties fail to hold in general for the restricted root space decomposition of a real Lie algebra; however, the dimension of Derq in the real case may be calculated on a example-by-example basis. 49 3.2 The real case In this section, g = gZ gS denotes a reductive real Lie algebra with center gZ and maximal semisimple ideal gS. q = gZ qS is a parabolic subalgebra of g, where qS = q\gS is a parabolic subalgebra of gS. We begin by proving the limited sense of the central theorem in the context of real Lie algebras. The proof will rely heavily on the complexification ?g of g, to which we will apply theorem 3.1. Afterwards, we consider the restricted root space decomposition of g and expand upon the central theorem. Theorem 3.3. For a parabolic subalgebra q = gZ qS of a reductive Lie algebra g = gZ gS over R, the derivation algebra Derq decomposes as the sum of ideals Derq = L adq, where L consists of all R-linear transformations on q mapping into qZ and mapping [q,q] to 0. Proof. We may assume without loss of generality that g is realized as a set of real matrices by Ado?s Theorem (proposition 2.5). We fix the following notation: i denotes the imaginary unit; ?g = g ?+ ig = (gZ ?+ igZ) (gS ?+ igS) denotes the complexification of g; q = gZ qS denotes a parabolic subalgebra of g; ?q = q ?+ iq = (gZ ?+ igZ) (qS ?+ iqS) is a parabolic subalgebra of ?g; and cgZ = gZ ?+ igZ = ?gZ denotes the center of ?g. Given a derivation D of q, we have a corresponding derivation ?D of ?q given by ?D(x + iy) = D(x)+ iD(y). As a derivation of ?q, ?D decomposes as ?D = L + ad(x + iy) with L 50 mapping ?q into ?gZ and mapping [?q, ?q] to 0 and with x, y2q. Note that L sends [q,q] to 0, since [q,q] [?q, ?q]. Let u2q. ?D stabilizes q, so we have D(u) = ?D(u) = L(u)+ ad(x + iy)(u) = L(u)+[x, u]+ i[y, u]2q. Now, L(u)2 ?gZ = gZ + igZ, so we may write L(u) = v1 + iv2 with v1, v2 2gZ. Then D(u) = v1 + iv2 +[x, u]+ i[y, u] = (v1 +[x, u])+ i(v2 +[y, u])2q so v2 + [y, u] = 0, and by the direct-sum decomposition q = gZ + [q,q], v2 = 0 and [y, u] = 0. In particular, we have L(u) = v1, so L maps q into gZ. Furthermore, since u was arbitrary and [y, u] = 0, we have y 2 gZ. Since y 2 gZ, we have for any arbitrary z = u + iv2g ad(x + iy)(z) = ad x(z)+ i ad y(z) = [x, u + iv]+ i[y, u + iv] = [x, u] [y, v]|{z} =0 +i([x, v]+[y, u]|{z} =0 ) = [x, u]+ i[x, v] = [x, u + iv] = ad x(z) thus we have ad(x + iy) = ad x. We now have D = Ljq +ad x with x2q and Ljq anR-linear transformation mapping q to gZ and [q,q] to 0, as desired. We have left to check that arbitrary R-linear maps 51 sending q to gZ and [q,q] to 0 are derivations, that L as described is an ideal of q, and that L and adq intersect trivially, the proofs of which are identical to the proofs given of claims 4, 5, and 6 of theorem 3.1, respectively. We will now examine the relationship between the direct sum decompositon of Derq and the restricted root space decomposition of g. Given a parabolic subalgebra q = gZ qS of a reductive real Lie algebra g = gZ gS, we may choose a restricted root space decomposition of g that is compatible with q in the sense that q is a standard parabolic subalgebra of g. We may then decompose q into the sum of q = gZ ?+ c ?+ [q,q] where c is an appropriately-chosen complimentary subspace, similar to the complex-like case. To achieve this decomposition, we rely on Langland?s decomposition of qS, described in chapter 2. We fix the following notation pertaining to the restricted root space decompo- sition of g: g = gZ ?+a ?+m ?+ a2Fga; D a base of F; D0 D corresponding to qS; F0 = F+[(F\SpanD0); and q = gZ ?+a ?+m ?+ g2F0 gg | {z } qS . Write Langland?s decomposition of qS: l = a ?+m ?+ g2F0\ F0gg and n = g2F0n F0 gg so that qS = l ?+n with l reductive and n nilpotent. Write c for the center of l and 52 lS for the unique semisimple ideal of l so that l = c ?+lS. Claim. [q,q] = lS ?+n Proof. Let x, y 2 q. We must show [x, y] 2 lS ?+ n. Without loss of generality, we may assume x, y2qS, since their projections onto gZ are lost upon applying bracket. Write x = xl + xn and y = yl + yn with xl, yl2l and xn, yn2n. Then [x, y] = [xl + xn, yl + yn] = [xl, yl]|{z} 2lS +[xl, yn]+[xn, yl]+[xn, yn]| {z } 2n 2lS ?+n since l is reductive and n is an ideal. Thus we arrive at the desired decomposition, q = gZ ?+l ?+n|{z} gS = gZ ?+c ?+lS ?+n = gZ ?+c ?+[q,q]. Theorem 3.4. For any root system F with respect to which q is a standard parabolic subalgebra, q decomposes as q = gZ ?+c ?+[q,q] and the ideal L of Derq consists of all R-linear transformation on q that map gZ ?+c to gZ and map [q,q] to 0, whereby Derq = Hom R (gZ ?+c,gZ) adq. Proof. The proof is essentially done. The majority is merely the description of the decom- position of q, already done above. We have left to show only that L = HomR(gZ +c,gZ), 53 which is obvious in light of the decomposition q = gZ ?+c ?+[q,q]. The sceptical reader is referred to figure 1.2, illustrating the block form of matrices in L. Because of the coarseness of the restricted root space decomposition, the dimension of c is not readily available in the real case, in contrast to the complex-like case. dimc may be calculated if given a specific real Lie algebra g and a specific standard parabolic subalgebra q. 3.3 Corollaries The following three corollaries represent extremal cases of the central theorem. Corol- lary 3.5 applies to arbitrary parabolic subalgebras of a semisimple Lie algebra (ie, the case gZ = 0). Corollaries 3.6 and 3.7 apply specifically to minimal parabolic subalgebras (ie, Borel subalgebras) and maximal parabolic subalgebras (ie, the entire Lie algebra g), re- spectively. Corollary 3.5. For a parabolic subalgebra q of a semisimple Lie algebra g over the field F (where F is complex-like or R), the derivation algebra Derq satisfies Derq = adq. Proof. By the central theorem, Derq = L adq, and since gZ = 0, we have L = 0. Corollary 3.5 was proven for Borel subalgebras of semisimle Lie algebras over an arbitrary field by Leger and Luks in [19]. Tolpygo found the same result for parabolic subalgebras of complex Lie algebras [29]. Our contribution is to include parabolic sub- algebras of real Lie algebras. 54 Corollary 3.6. For a Borel subalgebra b = gZ ?+ g0 ?+ a2F+ ga of the reductive Lie algebra g = gZ gS over the field F (where F is complex-like or R), the derivation algebra Derb satisfies Derb = Hom F (gZ ?+(g0)Z ,gZ) adb. Proof. Write bS = g0 ?+ a2F+ ga. Since a2F+ ga is clearly the nilpotent radical of bS, the Levi factor l = g0. Applying the central theorem gives the result. Farnsteiner proved corollary 3.6 over a complex-like field in [10]. As with corollary 3.5, our contribution is to extend this result to Borel subalgebras of real Lie algebras. Corollary 3.7. For a reductive Lie algebra g = gZ gS over the field F (where F is complex-like or R), the derivation algebra Derg satisfies Derg = gl(gZ) adg. Proof. gS is its own Levi factor. Being semisimple, the center of gS is trivial, so L consists of linear maps stabilizing gZ and sending gS = [g,g] to 0, which is isomorphic to gl(gZ). The final corollary provides a high-level abstract description of Derq useful for dim- ension-counting arguments. It is also satisfying on a theoretical level, since it relies on simple constructions that can be carried out on any Lie algebra, suggests that the result here for reductive Lie algebras might be generalized to larger classes of Lie algebras. Recall that q/[q,q] is the minimal abelian quotient of q. Since q = gZ ?+c ?+[q,q], we have gZ ?+c = q/[q,q]. Also, gZ = qZ, and adq = q/qZ, thus: Corollary 3.8. For a parabolic subalgebra q of a reductive Lie algebra g over a complex-like field or over R, we have Derq = Hom(q/[q,q],qZ) (q/qZ). Proof. Above. 55 Chapter 4 Zero Product Determined Derivation Algebras Let g be a Lie algebra over an arbitrary field F. g is called zero product determined if for each F-bilinear map j : g g !V into an F-vector space V, the condition that j(x, y) = 0 whenever [x, y] = 0 (4.1) implies the existence of an F-linear map f : g !V satisfying j(x, y) = f [x, y] for all x, y2g. (4.2) We will drop the reference to the base field F when the context is clear. However, the reader should remember that the condition that a Lie algebra is zero product determined is tied to the understood base field. Explicitly, a given Lie algebra g may consist of com- plex matrices and may be considered as either a real Lie algebra or a complex Lie algebra. It is conceivable that g may be zero product determined as a real Lie algebra, but not zero product determined as a complex Lie algebra, or vice versa. A few remarks. First, some terminology. A bilinear map satisfying property 4.1 is said to preserve zero products. Second, property 4.2 can be thought of as a map factoring property. The bracket [ , ] can be thought of as a bilinear map m defined by m : 8 >>< >>: g g !g (x, y)7![x, y] 56 Then the definition can be understood in terms of function composition as saying that the bilinear map j factors as the composition of a linear map f and the Lie bracket m; in symbols, g is zero product determined if an only if j(x, y) = 0 whenever m(x, y) = 0 implies9f , j = f m. In words, g is zero product determined if and only if every bilinear map j that preserves zero products factors through the bracket map m. Third remark: the setting as described above is not entirely desirable, since it com- bines notions of bilinearity and linearity, making the study problematic in certain settings. An equivalent definition can be phrased entirely in terms of linear maps by considering tensor products of vector spaces [5]. One may replace the Cartesian product g g with the tensor product g g without ambiguity, since linear maps on the tensor product g g are in one-to-one correspondence with bilinear maps on the Cartesian product g g [22, Ch. IX, ?8]. See figure 4.1 for a diagrammatic expression of the factorization j = f m in the tensor-product setting. Finally, the above definition (along with the reformulated definition in terms of ten- sor products) works equally well for associative algebras?such as the algebra Rn n of n by n real matrices with the usual matrix multiplication?and non-associative, non-Lie al- gebras if the bracket is replaced by an appropriate multiplication. In fact, the initial work on zero product determined algebras was done in the context of matrix algebras, consid- ering the standard associative matrix product, the non-associative Lie bracket, and the non-associative Jordan product [4]. For the remainder of this chapter, let K denote a complex-like field. 57 4.1 Zero product determined algebras Definition 4.1. A K-algebra is a pair (A, m) whereAis a K-vector space and m :A KA !A is an K-linear map. The image Im m is denotedA2. Definition 4.1 encompasses Lie algebras when m is defined by m(x y) = [x, y]. (We note in the Lie algebra case that A2 = [A,A].) The definition also includes associative algebras (eg, matrix algebras under the usual matrix product, Banach algebras) and other non-associative algebras, such as Jordan algebras. In the sequel, we will suppress the mention of the scalar field K when there is no danger of ambiguity. The definition of zero product determined as applied to algebras is originally due to Bre?ar, Gra?i?c, and S?nchez Ortega; it was motivated by applications to analysis on Ba- nach algebras [1, 4]. Definition 4.2, given below, is equivalent to that found in [4] but rephrased in terms of linear maps on tensor products rather than in terms of bilinear maps on Cartesian products [5]. A purely linear approach offers the advantage of con- sidering kernels and images of linear maps, alleviating certain difficulties found in the bilinear approach. Consider that for a bilinear map j, the image of j is not necessarily a subspace of the target vector space, and the notion of a kernel of j is non-existent. Definition 4.2. An algebra (A, m) is called zero product determined if for each vector space V and for each linear map j :A A !V, if j(a1 a2) = 0 whenever m(a1 a2) = 0, then there is a linear map f :A2 !B whereby j factors through m as j = f m. 58 A A j // m V A2 f 77oo ooo ooo Figure 4.1: Tensor definition of zero product determined The figure diagrammatically expresses the factorization j = f m. All maps depicted are linear. Existence of j and m is assumed a priori. f exists if j preserves zero products. We reference several results on zero product determined algebras that will be used in the sequel. Proposition 4.1 (Theorem 2.3 in [5]). An algebra (A, m) is zero product determined if and only if Ker m is generated by elementary tensors. Some terminology will be helpful for understand the statement of the theorem. Ele- ments ofA Aare called tensors. An elementary tensor is a member ofA Aof the form a1 a2 for a1, a2 2A. In general, an arbitrary tensor t2A Ais a linear combination of elementary tensors, ie, t = ni=1 a(i)1 a(i)2 for some positive integer n. WhileA Ais generated by elementary tensors through taking linear combinations, an arbitrary subspace of A A may fail to be generated by the elementary tensors it contains. In fact, there are non-trivial subspace of A A that contain no elementary tensors other than the zero tensor 0 = 0 0. Proposition 4.2 (Theorem 3.1 in [5]). Let I be an arbitrary set and for each i2 I let (Ai, mi) be an algebra. Consider the algebra direct sum (A, m) = (Li2IAi,Li2I mi). (A, m) is zero product determined if and only if each summand (Ai, mi) is zero product determined. 4.2 Parabolic subalgebras of reductive Lie algebras For this section we adopt several of the notational conventions of [33] for clarity. Let g = gZ gS be a finite-dimensional reductive Lie algebra over K with gZ abelian and gS semisimple. Let h be a Cartan subalgebra of gS and let F be the root system of gS relative 59 to h. Let D be a base of F, and denote the positive roots relative to D by F+. Then gS decomposes as gS = h ?+ b2Fgb. From each b 2 F+ we select a non-zero vector eb 2 gb. Then for each b there is a unique e b 2g b with the property that eb, e b, and hb = [eb, e b] span a subalgebra of gS isomorphic to sl2(K). Each gb for b 2F is spanned by the vector eb, and the vectors ha for a 2D form a basis of h. For each b 2F and each h 2h we have [h, eb] = b(h)eb, and for each pair b, g 2 F we have [eb, eg] 2 gb+g, from which we define Nb,g 2K by [eb, eg] = Nb,geb+g. Of any arbitrary parabolic subalgebra q of g, we assume without loss of generality that it is a standard parabolic subalgebra of g corresponding to some subset D0 D. More explicitly, the assumption is that that q = gZ h ?+ b2F0 gb , where F0 = F+[ (F\SpanD0). In [33], the authors prove ? with minor error ? that the parabolic subalgebras of the finite-dimensional simple Lie algebras over K are zero product determined. Specifically, lemma 2.2 of [33] contains the following unproven claim: For any a, g2F0 where a+g is a root, at least one of a+2g or 2a+g is not a root. The root system of the simple Lie algebra G2 provides a counterexample, as illustrated in figure 4.2. 6 ? 3 + QQ QQ Qs QQ QQ Qk - A A AAU A A AAK a g 2a+g a+ 2g Figure 4.2: G2 root system Both a+ 2b and 2a+ b are roots. 60 Lemma 2.2 of [33] does however provide a suitable base case for an induction argu- ment, which we give below. We are given a parabolic subalgebra q and a bilinear map j : q q ! V (V is some arbitrary vector space) of which we assume j(x, y) = 0 whenever [x, y] = 0. For each g2F0 we chose a dg 2h so that g(dg) = 1. Following Wang et al, we define a linear map f : [q,q] !V by f(ha) = j(ea, e a) for each a2D, f(eg) = j(dg, eg) for each g2G, and extending linearly. The basic theorem of [33] is to show that f([x, y]) = j(x, y) for all x, y 2q. Lemma 2.2 of [33] gives a special case that Wang et al. use to facilitate the proof of the basic theorem. Lemma 4.3 (Lemma 2.2 in [33]). For a, g2F0 , if a+g6= 0, then f([ea, eg]) = j(ea, eg). We will need Wang et al.?s Lemma 2.1 for the proof of lemma 4.3. We state it now for use later: for the proof we refer the reader to [33]. Proposition 4.4 (Lemma 2.1 in [33]). For all h2h and all b2F0, we have f([h, eb]) = f(h, eb). Proof of lemma 4.3. Let k be the largest integer such that ka + g is a root. The proof of Lemma 2.2 in [33] shows that the proposition holds in case k = 0, 1. We proceed by assuming for induction that the proposition hold for all pairs of roots b, d such that b + d6= 0 and b+d is not a root. Pick h2h so that g(h) = 0 and a(h) = 1. Let a0 = 1 and for each i from 1 to k let ai = 1i ai 1Na,(i 1)a+g. 61 Notice that iai ai 1Na,(i 1)a+b = 0 (4.3) for each i from 1 to k. Then we have " h ea, k i=0 aieia+g # = k i=0 ai h, eia+g k i=0 ai ea, eia+g = k i=0 ai(ia+g)(h)eia+g k i=0 aiNa,ia+ge(i+1)a+g = g(h)|{z} =0 eg + k i=1 iai ai 1Na,(i 1)a+g | {z } =0 by 4.3 eia+g ak [ea, eka+g]| {z } =0 = 0 and since j preserves zero products by assumption we arrive at j h ea, k i=0 aieia+g ! = 0. (4.4) By bilinearity of j we have 0 = j h ea, k i=0 aieia+g ! = k i=0 aij(h, eia+g) k i=0 aij(ea, eia+g), and by proposition 4.4 and the definition of ai and Na,(i 1)a+g aij(h, eia+g) = ai f([h, eia+g]) = ai f((ia+g)(h)eia+g) = ai f(ieia+g) = ai iN a,(i 1)a+g f([ea, e(i 1)a+g]) = ai 1 f([ea, e(i 1)a+g]). 62 for each i from 1 to k. Then equation 4.4 becomes 0 = a0 j(h, eg)| {z } =0 + k i=1 ai 1 f([ea, e(i 1)a+g]) k i=0 aij(ea, eia+g) = k 1 i=0 ai f([ea, eia+g]) k 1 i=0 aij(ea, eia+g) ak j(ea, eka+g)| {z } =0 = k 1 i=0 ai f([ea, eia+g]) j(ea, eia+g) . For i 1, applying the inductive hypothesis to the pair a, ia + g gives f([ea, eia+g]) j(ea, eia+g) = 0, so the sum reduces to the i = 0 term: 0 = f([ea, eg]) j(ea, eg), which is what we set out to show. We now state the Basic Theorem of Wang et al. for use later. The remainder of the proof is of course found in [33]. Proposition 4.5 (Basic Theorem in [33]). A parabolic subalgebra q of a simple Lie algebra g over K is zero product determined. The results of [5] allow us to generalize proposition 4.5 to reductive Lie algebras and their parabolic subalgebras. Lemma 4.6. If g is an abelian Lie algebra, then g is zero product determined. Proof. To say g is abelian is to say Ker m = g g, which is generated by elementary tensors x y 8x, y2g. By proposition 4.1, g is zero product determined. Theorem 4.7. Let q = gZ qS be a parabolic subalgebra of a reductive Lie algebra g = gZ gS over the field K. Then q is zero product determined. 63 Proof. gZ is zero product determined by lemma 4.6, qS is zero product determined by proposition 4.5. The direct sum q = gZ qS is zero product determined by proposition 4.2. In particular, the Borel subalgebra b = gZ ?+ b2F+ gb and the reductive Lie algebra g are zero product determined. 4.3 Derivations of parabolic subalgebras We now return to the original motivation of this dissertation. We apply the study of zero product determined algebras to the derivation algebra Derq of a parabolic subalge- bra q = gZ qS of the reductive Lie algebra g = gZ gS over the field K. Definition 4.3. Let n, k2Z 0. Denote by L(n, k) the subalgebra of gl(Cn+k) consisting of matrices whose (n + i)-th rows are zero for 1 i k. L(n, k) consists of complex matrices with block form 0 B@ n k n k 0 0 1 CA. As a Lie algebra, L(n, k) = mnn where: m = gl(Cn); n is abelian, consisting of n k ma- trices with trivial bracket; and the action of m on n is given by usual matrix multiplication ? la [x, y] = xy for all x2m, y2n. Notice that L(n, 0) = gl(Cn), so L(n, k) is zero product determined by 4.7 when k = 0. Lemma 4.8. L(n, k) is zero product determined. Proof. Without loss of generality, we assume k 1. Write L = L(n, k), and define m : L L // // [L, L] by m(x y) = [x, y]. By the rank-nullity theorem, we have dim Ker m = n4 + 2n3k + n2k2 n2 nk + 1. 64 We will exhibit a basis for Ker m consisting of elementary tensors. Notation as above, L = mnn. m is zero product determined by theorem 4.7. By proposition 4.1, Ker mjm m admits a basis consisting of n4 n2 + 1 elementary tensors of the form x y with x, y2m and [x, y] = 0. Since n is abelian, n is zero product determined by lemma 4.6, and proposition 4.1 provides n2k2 more elementary tensors of the form x y with x, y2n and [x, y] = 0. We require 2n3k nk more linearly independent elementary tensors in Ker m. Consider the 2n3k 2n2k tensors Ti,j,l,q = ei,j el,n+q 2m n and Ti,j,l,q = el,n+q ei,j 2n m for i, j, l n and q k with j6= l. Additionally, we have 2n2k 2nk tensors Si,j,q = ei,j ei,j+1 ej,n+q + ej+1,n+q 2m n and Si,j,q = ej,n+q + ej+1,n+q ei,j ei,j+1 2n m with i n, j n 1, and q k. Finally, we have nk tensors of the form R(i, q) = ei,i + ei,n+q ei,i + ei,n+q 2(m ?+n) (m ?+n) for i n and q k, giving the desired 2n3k nk elementary tensors in Ker m. We have left to show that these tensors are linearly independent. 65 Expanding Si,j,q we see that Si,j,q = ei,j ej,n+q ei,j+1 ej+1,n+q| {z } /2SpanfTi,j,l,qg + ei,j ej+1,n+q ei,j+1 ej,n+q| {z } 2SpanfTi,j,l,qg is not in the span of the Ti,j,l,q. A similar observation shows that Si,j,q is not in the span of the Ti,j,l,q tensors. Expanding R(i, q) we have R(i, q) = ei,i ei,i| {z } 2m m + ei,n+q ei,n+q| {z } 2n n + ei,i ei,n+q + ei,n+q ei,i| {z } 2m n ?+n m . Since ei,i ei,i and ei,n+q ei,n+q are in m m and n n, respectively, we may subtract them, and we have left to consider R0(i, q) = ei,i ei,n+q + ei,n+q ei,i. R0(i, q) is not in the span of Ti,j,l,q, Ti,j,l,q since individually ei,i ei,n+q and ei,n+q ei,i are not among the T i,j,l,q, Ti,j,l,q . Now, consider S i,i,q + Si,i,q if i < n (in case i = n we are done, since we require j n 1 in Si,j,q). We have Si,i,q + Si,i,q = ei,i ei,n+q + ei,n+q ei,i| {z } =R0(i,q) +T ei,i+1 ei+1,n+q + ei+1,n+q ei,i+1 with T 2Span Ti,j,l,q, Ti,j,l,q , so we have R0(i, q) = Si,i,q + Si,i,q T + ei,i+1 ei+1,n+q + ei+1,n+q ei,i+1. Write R00(i, q) = ei,i+1 ei+1,n+q + ei+1,n+q ei,i+1. If i = n 1 we are done. If i < n 1 we may reduce R00(i, q) using the same method as above, and so by induction we are done. Thus we have a basis for Ker m consisting of elementary tensors, and L(n, k) is zero product determined by proposition 4.1. 66 Recall from theorem 3.1 that q decomposes as q = gZ ?+c ?+[q,q] and Derq decomposes as Derq = L adq = Hom K (gZ ?+c,gZ) qS. Since qS is known to be zero product determined by proposition 4.5, we direct our at- tention to L = HomK(gZ ?+c,gZ), which is zero product determined in light of lemma 4.8. Recall that our study in chapter 3 on the direct sum decomposition of Derq was originally motivated by the question of whether the derivation algebras of certain Lie algebras were zero product determined. We conclude this chapter with the answer to this original question in the affirmative. Theorem 4.9. Let q be a parabolic subalgebra of a reductive Lie algebra g over a complex-like field K. The derivation algebra Derq is zero product determined. Proof. We begin with the decomposition Derq = L adq established by theorem 3.1. We have adq = qS, and qS is zero product determined by proposition 4.5. Furthermore, L is zero product determined. To verify this, write n = dimgZ and k = dimc, and observe that L = L(n, k). By lemma 4.8, L is zero product determined. By proposition 4.2, Derq, as the direct sum of zero product determined Lie algebras, is zero product determined. 67 Chapter 5 Examples and Future Research We close this dissertation by taking note of directions that future research could take and how such research would fit into the existing body of results, and by providing worked examples and tabular data for reductive Lie algebras of types A5, G2, and F4. 5.1 Examples We provide an algorithmic method for computing the center lZ of the Levi factor in the Langland?s decomposition of a parabolic subalgebra corresponding to any given subset of the base D of the root system. We then enumerate all standard parabolic subalg- ebras and give the dimensions of L, qS (which, recall, is isomorphic to adq), and Derq in tabular form. 5.1.1 Type A5 Let g = Cn gS where gS = sl(C6). gS has the root space decomposition gS = h ?+ i6=j Cei,j where h consists of traceless diagonal 6 6 complex matrices. Chose D =fa1, ..., a5gas a base where gai = Cei,i+1. Then F+ = n 5i=1 aiai ai 2f0, 1g o and F = F+[ F. Write xi = ei,i+1, yi = ei+1,i, and hi = [xi, yi] = ei,i ei+1,i+1. For each i, let ti be the coroot dual to ai, so ai(tj) = di,j. T = ft1, ..., t5gis a basis for h. Partial multiplication table for g in terms ofH=fh1, ..., h5gandT are provided in tables 5.1 and 5.2 respectively. 68 x1 x2 x3 x4 x5 h1 2x1 x2 0 0 0 h2 x1 2x2 x3 0 0 h3 0 1x2 2x3 x4 0 h4 0 0 x3 2x4 x4 h5 0 0 0 x4 2x5 Table 5.1: Partial multiplication table for sl(C6) in terms ofH x1 x2 x3 x4 x5 t1 x1 0 0 0 0 t2 0 x2 0 0 0 t3 0 0 x3 0 0 t4 0 0 0 x4 0 t5 0 0 0 0 x5 Table 5.2: Partial multiplication table for sl(C6) in terms ofT For any D0 D with corresponding parabolic subalgebra q = gZ ?+h ?+ b2F0gb, we make three observations. First, the derived algebra [q,q] is determined by D0 as [q,q] = Span hi ai 2D0 ?+ b2F0 gb. Second, the center lZ of the Levi factor l is given by lZ = Span ti ai 2DnD0 . Third, the matrix whose columns are the members of T written as vectors in terms of the basisHis the inverse of the transpose of the Cartan matrix of g. Figure 5.1 gives the Cartan matrix and the inverse transpose of g, and table 5.3 gives members ofT in terms ofHand as matrices. Utilizing the three above observations allows one to explicitly compute a basis for the ideal L of Derg. Table 5.4 contains data on for all standard parabolic subalgebras of g with respect to h. 69 ti ti in terms ofH ti as a diagonal matrix t1 (5/6, 2/3, 1/2, 1/3, 1/6) diag(5/6, 1/6, 1/6, 1/6, 1/6, 1/6) t2 (2/3, 4/3, 1, 2/3, 1/3) diag(2/3, 2/3, 1/3, 1/3, 1/3, 1/3) t3 (1/2, 1, 3/2, 1, 1/2) diag(1/2, 1/2, 1/2, 1/2, 1/2, 1/2) t4 (1/3, 2/3, 1, 4/3, 2/3) diag(1/3, 1/3, 1/3, 1/3, 2/3, 2/3) t5 (1/6, 1/3, 1/2, 2/3, 5/6) diag(1/6, 1/6, 1/6, 1/6, 1/6, 5/6) Table 5.3:T in terms ofHand as matrices A = 2 66 66 4 2 1 0 0 0 1 2 1 0 0 0 1 2 1 0 0 0 1 2 1 0 0 0 1 2 3 77 77 5 (AT) 1 = 2 66 66 4 5/6 2/3 1/2 1/3 1/6 2/3 4/3 1 2/3 1/3 1/2 1 3/2 1 1/2 1/3 2/3 1 4/3 2/3 1/6 1/3 1/2 2/3 5/6 3 77 77 5 Figure 5.1: Cartan matrix and transpose inverse for Type A5 5.1.2 Type G2 Let g = Cn gS where gS is simple of type G2. The same observations in the previous example apply to any parabolic subalgebra corresponding to a D0 D. In particular, lZ = Span tiai 2DnD0. Figure 5.2 gives the Cartan matrix for Type G2 and gives the inverse transpose, whose columns are ti in terms of the hi. Table 5.5 gives data for all the standard parabolic subalgebras of g. 5.1.3 Type F4 Let g = Cn gS where gS is simple of type F4. Again, to any parabolic subalgebra corresponding to a D0 D lZ = Span tiai 2DnD0. Figure 5.3 gives the Cartan matrix for Type F4 and gives the inverse transpose, whose columns are ti in terms of the hi. Table 5.6 gives data for all the standard parabolic subalgebras of g. A = 2 3 1 2 (AT) 1 = 2 1 3 2 Figure 5.2: Cartan matrix and transpose inverse for Type G2 70 D0 dimlZ dimL dimqS dim Derq ? 5 n2 + 5n 20 n2 + 5n + 20 10000 4 n2 + 4n 21 n2 + 4n + 21 01000 4 n2 + 4n 21 n2 + 4n + 21 00100 4 n2 + 4n 21 n2 + 4n + 21 00010 4 n2 + 4n 21 n2 + 4n + 21 00001 4 n2 + 4n 21 n2 + 4n + 21 11000 3 n2 + 3n 23 n2 + 3n + 23 10100 3 n2 + 3n 22 n2 + 3n + 22 10010 3 n2 + 3n 22 n2 + 3n + 22 10001 3 n2 + 3n 22 n2 + 3n + 22 01100 3 n2 + 3n 23 n2 + 3n + 23 01010 3 n2 + 3n 22 n2 + 3n + 22 01001 3 n2 + 3n 22 n2 + 3n + 22 00110 3 n2 + 3n 23 n2 + 3n + 23 00101 3 n2 + 3n 22 n2 + 3n + 22 00011 3 n2 + 3n 23 n2 + 3n + 23 11100 2 n2 + 2n 26 n2 + 2n + 26 11010 2 n2 + 2n 24 n2 + 2n + 24 11001 2 n2 + 2n 24 n2 + 2n + 24 10110 2 n2 + 2n 24 n2 + 2n + 24 10101 2 n2 + 2n 23 n2 + 2n + 23 10011 2 n2 + 2n 24 n2 + 2n + 24 01110 2 n2 + 2n 26 n2 + 2n + 26 01101 2 n2 + 2n 24 n2 + 2n + 24 01011 2 n2 + 2n 24 n2 + 2n + 24 00111 2 n2 + 2n 26 n2 + 2n + 26 11110 1 n2 + n 30 n2 + n + 30 11101 1 n2 + n 27 n2 + n + 27 11011 1 n2 + n 26 n2 + n + 26 10111 1 n2 + n 27 n2 + n + 27 01111 1 n2 + n 30 n2 + n + 30 D 0 n2 35 n2 + 35 Table 5.4: Parabolic subalgebras of type A5 D0 lZ dimL dimqS dim Derq ? h n2 + 2n 8 n2 + 2n + 8 fa1g Spanfh1 + 2h2g n2 + n 9 n2 + n + 9 fa2g Spanf2h1 + 3h2g n2 + n 9 n2 + n + 9 D 0 n2 14 n2 + 14 Table 5.5: Parabolic subalgebras of type G2 71 A = 2 66 4 2 1 0 0 1 2 2 0 0 1 2 1 0 0 1 2 3 77 5 (AT) 1 = 2 66 4 2 3 2 1 3 6 4 2 4 8 6 3 2 4 3 2 3 77 5 Figure 5.3: Cartan matrix and transpose inverse for Type F4 D0 dimlZ dimL dimqS dim Derq ? 4 n2 + 4n 28 n2 + 4n + 28 1000 3 n2 + 3n 29 n2 + 3n + 29 0100 3 n2 + 3n 29 n2 + 3n + 29 0010 3 n2 + 3n 29 n2 + 3n + 29 0001 3 n2 + 3n 29 n2 + 3n + 29 1100 2 n2 + 2n 31 n2 + 2n + 31 1010 2 n2 + 2n 30 n2 + 2n + 30 1001 2 n2 + 2n 30 n2 + 2n + 30 0110 2 n2 + 2n 32 n2 + 2n + 32 0101 2 n2 + 2n 30 n2 + 2n + 30 0011 2 n2 + 2n 31 n2 + 2n + 31 1110 1 n2 + n 37 n2 + n + 37 1101 1 n2 + n 32 n2 + n + 32 1011 1 n2 + n 32 n2 + n + 32 0111 1 n2 + n 37 n2 + n + 37 D 0 n2 54 n2 + 54 Table 5.6: Parabolic subalgebras of type F4 72 5.2 Directions for future research The theorems of chapter 4 apply primarily to reductive Lie algebras over complex- like fields. An immediate extension would be to prove these results for reductive Lie algebras over R. Our results in chapter 4 extend work carried out by Wang et al. in [33], where the authors prove that the parabolic subalgebras of a simple Lie algebra over a complex-like field are zero product determined. The arguments employed by Wang et al. do not appear to be easily extended to the real case, as the computational method employed relies on the fact that root spaces are one-dimensional, where restricted root spaces can be arbitrarily large in dimension. An abstract argument, however, similar to our proof of theorem 3.3 from theorem 3.1 might produce the desired result. Along the same lines, we may extend the class of Lie algebras under consideration by including Lie algebras over prime-characteristic fields [19, 26] or over general commu- tative rings [23, 30, 31]. Alternatively, we may consider infinite-dimensional Lie algebras. Kac-Moody algebras are infinite-dimensional generalizations of the (finite-dimensional) semisimple Lie algebras, and they share many of the properties of semisimple Lie alge- bras especially as they relate to root space decomposition [16]. Farnsteiner in 1988 inves- tigated the derivations of Borel subalgebras of Kac-Moody algebras, and perhaps similar techniques can be employed to extend these results to parabolic subalgebras [10]. The methods we employ in our investigation have several noteworthy precedents in the literature. Recall, for instance, the discussion in chapter 1 of the work of Jacobson in 1955 in [14] and the related work by Dixmier and Lister in 1957 in [9]. Dixmier and Lister show that a converse to a result proved by Jacobson is not possible by constructing an ex- ample of a nilpotent Lie algebra and explicitly describing its derivation algebra. Dixmier and Lister employ methods in their concrete example that mirror the abstract methods we use in this dissertation. The interesting point is this: parabolic subalgebras are never nilpotent. This suggests that perhaps the methods empolyed here may be extended to a much wider classes of Lie algebras. 73 In contrast, we may consider the methods of Leger and Luks in [19] and the meth- ods of Tolpygo in [29]. In these papers, the authors prove special cases of our results, though they use completely different methods. Leger?s and Luks?s results imply that all derivations of a Borel subalgebra of a simple Lie algebra are inner (over any field with characteristic not 2) [19], and similarly Tolpygo?s results (applicable specifically over the complex field) imply that all derivations of a parabolic subalgebra of a semisimple Lie algebra are inner [29]. In fact, these results are more general and stated in the language of cohomology: The authors prove that all cohomology group Hn(g,g) are trivial for their respective classes of Lie algebras g under consideration [19, 29]. Very briefly, cohomology groups are computable invariants of a Lie algebra that pro- vide information about the Lie algebra under consideration. (For context, the reader is reminded that the familiar Calculus derivative f0 of a real-valued function f is a com- putable invariant that provides information about f .) For instance, the first cohomology group H1(g;g) of a Lie algebra g satisfies the isomorphism H1(g;g) = Derg/ adg. From this isomorphism, it follows that H1(g;g) = 0 implies that all derivations of g are inner. A further application of cohomology is to extensions of a Lie algebra b by an ideal a. We have the isomorphism H2(b;a) = Ext(b;a) meaning that the second cohomology group H2(b;a) parametrizes the set of all possible extensions of b by a (cf. definition 2.11). If we happen to know that H2(b;a) = 0, then we know that the only extension of b by a is the trivial extension defined by the action 8b 2b,8a 2a, b a = 0. Such an action results in a component-wise bracket rule, so the extensions is in fact the Lie algebra direct sum b a. 74 In light of these two isomorphisms, the language and methods of cohomology pro- vide a strong framework for discovering structural properties of Derg as they relate to properties of g. Our results on derivations apply to reductive Lie algebras, trivial ex- tensions of semisimple Lie algebras by an abelian Lie algebra. Cohomology might be employed to study the derivations of general extensions of Lie algebras. Our results on direct sums of zero product determined algebras might likewise be generalized to exten- sions of algebras. Our present results, in turn, can enrich the study of cohomology by providing further examples of classes of Lie algebras that exhibit non-trivial first coho- mology groups. Another direction for extending our results is to generalize the notion of derivation itself. A straightforward way of doing this is to drop the requirement that a derivation be linear, an approach studied in by Chen and Wang in [6] and [7]. The authors use the term nonlinear map satisfying derivability for a not-necessarily-linear map D : g !g satisfying 8x, y2g D [x, y] = D(x), y + x, D(y) . (5.1) Alternatively, one can generalize the notion of a derivation by relaxing the product rule (equation 5.1), studying linear maps D : g !g satisfying the weaker condition 8x, y, z2g D [x, y], z = h D(x), y , z i + h x, D(y) , z i + h x, y , D(z) i . (5.2) Such a map is called a Lie triple derivation, and these maps are studied in [20], [35], and [32] among others. The two approaches can be combined, studying maps D that are not neces- sarily linear and satisfy condition 5.2 rather than condition 5.1. Chen and Wang take this approach in [8], naming such a map a nonlinear Lie triple derivation. Figure 5.4 illustrates the logical connection between the various types of derivation-like maps considered. We offer a brief summary of the results in [6] and [8]. In [6], Chen and Wang study non-linear maps satisfying derivability on parabolic subalgebras of simple Lie algebras 75 Derivations Non-linear triple derivations Triple derivations Non-linear maps satisfying derivability =) =) =) =) Figure 5.4: Logical relations among various types of derivation-like maps over a complex-like field. The authors show that any such map is the sum of an inner derivation and what the authors call an additive quasi-derivation (a map that, notably, fails to be homogeneous). As an aside, this gives an alternate proof that Der(q) = ad(q) when q is a parabolic subalgebra of a simple Lie algebra. In [8], the same authors study non-linear Lie triple derivations in the same setting, parabolic subalgebras of a simple Lie algebra g over a complex-like field. What is worth noting, though, is that their result is ex- actly the same: a non-linear Lie triple derivation is the sum of an inner derivation and an additive quasi-derivation. In other words, the weaker assumption of requiring condition 5.2 rather than condition 5.1 resulted in no new maps ? the non-linear Lie triple deriva- tions and the non-linear maps satisfying derivability on a parabolic subalgebra exactly coincide in case g is simple. The results of Chen and Wang motivate the following questions: Are there non-linear triple derivations that are not non-linear maps satisfying derivability? If so, what classes of Lie algebras must we consider in order to differentiate between the two types of maps? It would be interesting to extend these results to parabolic subalgebras of reductive Lie algebras for a number of reasons. Considering derivations of reductive algebras has pro- vided examples of derivations that are non inner ? a sort of non-triviality result about 76 outer derivations. In parallel, considering non-linear maps satisfying derivability and non-linear triple derivations of parabolic subalgebras of reductive algebras could perhaps lead to examples of non-linear triple derivations that are not non-linear maps satisfying derivability. A final vehicle for future research that we will discuss deals with the abstract form of the decomposition of Derq established by theorems 3.1 and 3.3. If we denote by g a parabolic subalgebra, we have that the derivation algebra Derg decomposes as Derg = Hom(g/[g,g],gZ) adg. (5.3) by corollary 3.8. The constructions adg, gZ, g/[g,g], and Hom(g/[g,g],gZ) can be carried out for any Lie algebra g, motivating the following question: for which Lie algebras g does isomorphism 5.3 hold? 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