arXiv subjects
Publications and source records attributed to Daniel Brice.
The class of matrix algebras on a ladder $\mathcal{L}$ generalizes the class of block upper triangular matrix algebras. It was previously shown that the matrix algebra on a ladder $\mathcal{L}$ is zero product determined under matrix multiplication. In this article, we show that the matrix algebra on a one-step ladder is zero product determined under the Lie bracket.
Let $\mathfrak{g}$ be a reductive Lie algebra over an algebraically closed, characteristic zero field or over $\mathbb{R}$. Let $\mathfrak{q}$ be a parabolic subalgebra of $\mathfrak{g}$. We characterize the derivations of $\mathfrak{q}$ by decomposing the derivation algebra as the direct sum of two ideals: one of which being the image of the adjoint representation and the other consisting of all linear transformations on $\mathfrak{q}$ that map into the center of $\mathfrak{q}$ and map the derived algebra of $\mathfrak{q}$ to $0$.
We reformulate the definition of a zero product determined algebra in terms of tensor products and obtain necessary and sufficient conditions for an algebra to be zero product determined. These conditions allow us to prove that the direct sum \bigoplus_{i \in I} A_i of algebras for any index set I is zero product determined if and only if each of the component algebras A_i is zero product determined. As an application, every parabolic subalgebra of a finite-dimensional reductive Lie algebra, over an algebraically-closed field of characteristic zero, is zero product determined. In particular, every such reductive Lie algebra is zero product determined.