Minimal Faithful Representation of the Heisenberg Lie Algebra with Abelian Factor

For a finite dimensional Lie algebra g over a field k of characteristic zero, the mu-function (respectively mu(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) circle plus a(n)) and mu(nil)(h(m) circle plus a(n)) for all m, n is an element of N. We also give a faithful representation and faithful nilrepresentation of h(m) circle plus a(n) of minimal dimension for all m, n is an element of N.