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

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 $\mathfrak{gl}(n,\K)$.