Generalized Heisenberg Algebras and Toroidal Lie Algebras

Abstract

In this article we provide two kinds of infinite presentations of toroidal Lie algebras. At first we define generalized Heisenberg algebras and prove that each toroidal Lie algebra is an amalgamation of a simple Lie algebra and a generalized Heisenberg algebra in the sense of Saito and Yoshii. This is one kind of presentations of toroidal Lie algebras given by the generators of generalized Heisenberg algebras and the Chevalley generators of simple Lie algebras with certain amalgamation relations. Secondly by using the generalized Chevalley generators, we give another kind of presentations. These two kinds of presentations are different from those given by Moody, Eswara Rao and Yokonuma.