Cohomologie de Hochschild et crochet de Gerstenhaber d'une famille de sous algebres de l'algebre de Weyl

A. SOLOTAR

Conferencia; Seminaire GAAO; 2019

Universite de Clermont-Auvergne

For each nonzero h∈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 the Hochschild cohomology HH∙(Ah) over a field of arbitrary characteristic. In case F has positive characteristic, the center of Ah is nontrivial and we describe HH∙(Ah) as a module over its center. The most interesting results occur when F has 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(A)-module.