For each nonzero , where is a field, let be the unital associative algebra generated by elements , satisfying the relation . This gives a parametric family of subalgebras of the Weyl algebra , containing many well-known algebras which have previously been studied independently. In this paper, we give a full description of the Hochschild cohomology over a field of an arbitrary characteristic. In case has a positive characteristic, the center of is nontrivial and we describe as a module over . The most interesting results occur when has a characteristic . In this case, we describe as a module over the Lie algebra and find that this action is closely related to the intermediate series modules over the Virasoro algebra. We also determine when is a semisimple -module.
Cite this article
Samuel A. Lopes, Andrea Solotar, Lie structure on the Hochschild cohomology of a family of subalgebras of the Weyl algebra. J. Noncommut. Geom. 15 (2021), no. 4, pp. 1373–1407DOI 10.4171/JNCG/439