Simple Poisson modules over certain polynomial Poisson algebras

Abstract

In this paper, we study a Poisson algebra A constructed from the C-algebra T with generators x, y, z and t±1 and subject to the relations as follows: xy − t−2yx = (1− t−2)z, yz − t−2zy = (1− t−2)x, zx− t−2xz = (1− t−2)y, and xt = tx, yt = ty, zt = tz, tt−1 = 1 = t−1t, where a, b, c ∈ C \\ {0} and t is a central element of T . The main result shows that there are five d-dimensional simple modules over Poisson algebra A for any d ≥ 1. Mathematics Subject Classifications: 17B63, 16S10