Minimal Faithful Representation of the Heisenberg Lie Algebra with Abelian Factor

N. E. Rojas

Journal of Lie theory, Tome 23 (2013) no. 4, pp. 1105-1114

Résumé

\def\a{{\frak a}} \def\g{{\frak g}} \def\h{{\frak h}} \def\k{{\frak k}} \def\N{{\Bbb N}} For a finite dimensional Lie algebra $\g$ over a field $\k$ of characteristic zero, the $\mu$-function (respectively $\mu_{\rm{nil}}$-function) is defined to be the minimal dimension of $V$ such that $\g$ admits a faithful representation (respectively a faithful nilrepresentation) on $V$. Let $\h_m$ be the Heisenberg Lie algebra of dimension $2m + 1$ and let $\a_n$ be the abelian Lie algebra of dimension $n$. The aim of this paper is to compute $\mu(\h_m \oplus \a_n)$ and $\mu_{\rm{nil}}(\h_m \oplus \a_n)$ for all $m,n \in \N$. We also give a faithful representation and faithful nilrepresentation of $\h_m \oplus \a_n$ of minimal dimension for all $m,n \in \N$.

Détail

Classification : 17B10, 17B30, 20C40
Mots-clés : Nilpotent Lie algebras, Heisenberg Lie algebra, Ado's Theorem, minimal faithful representation, nilrepresentation