On the derivation algebras of parabolic Lie algebras with applications to zero product determined algebras

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 φ: g × g → V that satisfies φ(x, y) = 0 whenever [x, y] = 0 there is a linear map f: [g, g] → V such that φ(x, y) = f([x, y]) for all x, y ∈ g. 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 ∈ g. Der g 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 Der q. Der q decomposes as the direct sum of ideals Der q = L ⊕ ad q, 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 Der q 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 applying the main results to providing tabular data for parabolic subalgebras of reductive Lie algebras of types A5, G2, and F4.