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

Bechir Dali (1)

1 Département de mathématiques. Unité de recherche analyse harmonique

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

Mots clés: Lie algebras; Matrices; Extension; Contraction

Domaines: Théorie des représentations [math.RT]

Bechir Dali. Linear family of Lie brackets on the space of matrices $Mat(n\times m,\K)$ and Ado's Theorem. 2008. ⟨hal-00295135⟩