On the Lie Algebra Associated to the Canonical Matrices

Abstract

For each fixed complex matrix M, the solution space to the matrix equation XM + MX^T = 0 is a Lie algebra denoted by $\mathfrak{g}(n,M,\mathbb{C})$. We study the basic structure of $\mathfrak{g}(n,M,\mathbb{C})$ when M are the canonical matrices for congruence, i.e. M = J_n(0), Γ_n,H_2n(λ). We show that $\mathfrak{g}(n,J_{n}(0),\mathbb{C})$ (n even) and $\mathfrak{g}(n,\Gamma_{n},\mathbb{C})$ are abelian Lie algebras; and $\mathfrak{g}(n,J_{n}(0),\mathbb{C})$ (n > 1 odd) is a non-nilpotent solvable Lie algebra. For $\mathfrak{g}(2n,H_{2n}(\lambda),\mathbb{C})$, we determine its basis, radical and Levi subalgebra, and show that its radical is nilpotent.