Lie structure on the Hochschild cohomology of a family of subalgebras of the Weyl algebra

Abstract

For each nonzero h 2 F [x], where F is a field, let Ah be the unital associative algebra generated by elements x, y, satisfying the relation yx - xy = h. This gives a parametric family of subalgebras of the Weyl algebra A1, containing many well-known algebras which have previously been studied independently. In this paper, we give a full description of the Hochschild cohomology HH'(Ah) over a field of an arbitrary characteristic. In case F has a positive characteristic, the center Z(Ah) of Ah is nontrivial and we describe HH'(Ah) as a module over Z(Ah). The most interesting results occur when F has a characteristic 0. In this case, we describe HH'(Ah) as a module over the Lie algebra HH1(Ah) and find that this action is closely related to the intermediate series modules over the Virasoro algebra. We also determine when HH'(Ah) is a semisimple HH1(Ah)-module.