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 nu(n, m) for H(n,m). Considering the Poisson subalgebra K over circle as 1-dimensional central extension of HBAR(n, m), 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).