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

2019/03/04 by Samuel A. Lopes, Lopes, Samuel A., Andréa Solotar +1
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.1903.01226

openalex publication_date 2019/03/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For each nonzero h∈ \mathbbF[x], where \mathbbF is a field, let A_h 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^\bullet(A_h) over a field of arbitrary characteristic. In case \mathbbF has positive characteristic, the center of A_h is nontrivial and we describe HH^\bullet(A_h) as a module over its center. The most interesting results occur when \mathbbF has characteristic 0. In this case, we describe HH^\bullet(A_h) as a module over the Lie algebra HH¹(A_h) and find that this action is closely related to the intermediate series modules over the Virasoro algebra. We also determine when HH^\bullet(A_h) is a semisimple HH¹(A)-module.

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