On Properties of the (2n+1)-Dimensional Heisenberg Lie Algebra

Abstract

In the present paper, we study some properties of the Heisenberg Lie algebra of dimension . The main purpose of this research is to construct a real Frobenius Lie algebra from the Heisenberg Lie algebra of dimension . To achieve this, we exhibit how to compute the derivation of the Heisenberg Lie algebra by following Oom’s result. In this research, we use a literature review method to some related papers corresponding to a derivation of a Lie algebra, Frobenius Lie algebras, and Plancherel measure. Determining a conjecture of a real Frobenius Lie algebra is obtained. As the main result, we prove that conjecture. Namely, for the given the Heisenberg Lie algebra, there exists a commutative subalgebra of dimension one such that its semi direct sum is a real Frobenius Lie algebra of dimension . Futhermore, in the notion of the Lie group of the Heisenberg Lie algebra which is called the Heisenberg Lie group, we compute the generalized character of its group and we determine the Plancherel measure of the unitary dual of the Heisenberg Lie group. As our contributions, we complete some examples of Frobenius Lie algebras obtained from a nilpotent Lie algebra and we also give alternative computations to find the Plancherel measure of the Heisenberg Lie group.