Linear family of Lie brackets on the space of matrices $Mat(n\times m,\K)$ and Ado's Theorem

TL;DR

This paper classifies linear families of Lie brackets on matrix spaces, extends Ado's Theorem, and explores representations and contractions of related Lie algebras, revealing limitations on faithful representations of certain Heisenberg algebras.

Contribution

It provides a classification of Lie brackets on matrix spaces, offers an analogue of Ado's Theorem, and analyzes representation limitations and contractions of matrix Lie algebras.

Findings

Classified linear families of Lie brackets on matrix spaces.

Established an analogue of Ado's Theorem for these structures.

Proved non-existence of faithful representations of certain Heisenberg algebras.

Abstract

In this paper we classify a linear family of Lie brackets on the space of rectangular matrices $Mat(n\times m,\K)$ and we give an analogue of the Ado's Theorem. We give also a similar classification on the algebra of the square matrices $Mat(n, \K)$ and as a consequence, we prove that we can't built a faithful representation of the $(2n+1)$-dimensional Heisenberg Lie algebra $\mathfrak{H}_n$ in a vector space $V$ with $\dim V\leq n+1$. Finally, we prove that in the case of the algebra of square matrices $Mat(n,\K)$, the corresponding Lie algebras structures are a contraction of the canonical Lie algebra $\mathfrak{gl}(n,\K)$.