My research interests include Lie algebras and representation theory, linear algebra and matrix theory, and related topics. Specifically, I’ve done work in derivations of Lie algebras and zero product determined algebras. This page gives a high-level overview of these topics. Please see my curriculum vitae for details.

## Derivations of Parabolic Lie Algebras

A derivation on a Lie algebra $$L$$ is a map $$f: L \to L$$ satisfying

$f\big([x,y]\big) = \big[f(x), y\big] + \big[x, f(y)\big]$

for all $$x, y \in L$$.

I proved that the derivations of a parabolic Lie algebra (a Lie algebra that is realized as a parabolic subalgebras of some reductive Lie algebra) are the sums of inner derivations and linear maps into the center that kill the derived algebra.

Explicitly, given a parabolic Lie algebra $$L$$, the derivations algebra $$\mathrm{Der} (L)$$ decomposes as the direct sum of ideals

$\mathrm{Der} (L) = \mathrm{ad} (L) \oplus \mathcal L$

where

$\mathcal L = \left\{ f: L \to L ; f(L) \subseteq Z(L), f([L,L]) = 0 \right\}.$

These results appeared in the Journal of Lie Theory in 2017 (article).

## Zero Product Determined Algebras

An algebra $$A$$ is zero product determined if for each bilinear map $$\varphi: A \times A \to V$$ satisfying

$\varphi(x, y) = 0 \text{ whenever } xy=0$

there is a linear map $$f: A^2 \to A$$ such that

$\varphi(x, y) = f(xy)$

for all $$x, y \in A$$.

Huajun Huang and I wrote a paper on general zero product determined algebras that appeared in Linear and Multilinear Algebra in 2015, and I’m currently working on a second paper concerning specifically zero product determined Lie algebras. I’ll go into more details some time in the future (still working as of Jan 2017).

## List of Publications

A selection of my peer-reviewed and submitted research articles.

## List of Presentations

Selected conference and seminar presentations.