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

  • Samuel A. Lopes

    Universidade do Porto, Portugal
  • Andrea Solotar

    Universidad de Buenos Aires, Argentina
Lie structure on the Hochschild cohomology of a family of subalgebras of the Weyl algebra cover
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Abstract

For each nonzero hF[x]h\in\mathbb{F}[x], where F\mathbb{F} is a field, let Ah\mathsf{A}_h be the unital associative algebra generated by elements x,yx,y, satisfying the relation yxxy=hyx-xy=h. This gives a parametric family of subalgebras of the Weyl algebra A1\mathsf{A}_1, 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)\operatorname{\mathsf{HH}}^{\bullet}(\mathsf{A}_h) over a field of an arbitrary characteristic. In case F\mathbb{F} has a positive characteristic, the center Z(Ah)\mathsf{Z}(\mathsf{A}_{h}) of Ah\mathsf{A}_h is nontrivial and we describe HH(Ah)\operatorname{\mathsf{HH}}^\bullet(\mathsf{A}_h) as a module over Z(Ah)\mathsf{Z}(\mathsf{A}_{h}). The most interesting results occur when F\mathbb{F} has a characteristic 00. In this case, we describe HH(Ah)\operatorname{\mathsf{HH}}^\bullet(\mathsf{A}_h) as a module over the Lie algebra HH1(Ah)\operatorname{\mathsf{HH}}^1(\mathsf{A}_h) and find that this action is closely related to the intermediate series modules over the Virasoro algebra. We also determine when HH(Ah)\operatorname{\mathsf{HH}}^\bullet(\mathsf{A}_h) is a semisimple HH1(Ah)\operatorname{\mathsf{HH}}^1(\mathsf{A}_h)-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. 1373–1407

DOI 10.4171/JNCG/439