The graded modules for the graded contact cartan algebras

In this paper we investigate the graded modules for the graded contact Cartan algebras K(n, m) and K(n). For a canonical basis of uPTG module, we derive a commutator formula and then realize Shen’s mixed product module in uPTG module V(n, m) for 0 H(n, m). Considering the Poisson subalgebra [formula omitted] as 1-dimensional central extension of [formula omitted]([formula omitted], [formula omitted]), we describe the irreducible PTG modules for K(n, m) and K(n), respectively. In particular, for arbitrary K(n,m), we recover Holmes’ work for K(n,1).