The hom-associative Weyl algebras in prime characteristic

Per Back,Johan Richter

doi:10.24330/ieja.1058430

Abstract

We introduce the first hom-associative Weyl algebras over a field of prime characteristic as a generalization of the first associative Weyl algebra in prime characteristic. First, we study properties of hom-associative algebras constructed from associative algebras by a general ``twisting'' procedure. Then, with the help of these results, we determine the commuter, center, nuclei, and set of derivations of the first hom-associative Weyl algebras. We also classify them up to isomorphism, and show, among other things, that all nonzero endomorphisms on them are injective, but not surjective. Last, we show that they can be described as a multi-parameter formal hom-associative deformation of the first associative Weyl algebra, and that this deformation induces a multi-parameter formal hom-Lie deformation of the corresponding Lie algebra, when using the commutator as bracket.

Highlights

Hom-associative algebras originate with the introduction of hom-Lie algebras, the latter a class of algebras defined by Hartwig, Larsson, and Silvestrov [10] to describe deformations of Lie algebras obeying a generalized Jacobi identity twisted by a homomorphism; the name
Hom-associative algebras, introduced by Makhlouf and Silvestrov [14], play the same role as associative algebras do for Lie algebras; equipping a hom-associative algebra with the commutator as bracket give rise to a hom-Lie algebra
In a hom-associative algebra, the associativity condition is twisted by a homomorphism, to the twisting of the Jacobi identity in hom-Lie algebras

Summary

Introduction

Hom-associative algebras originate with the introduction of hom-Lie algebras, the latter a class of algebras defined by Hartwig, Larsson, and Silvestrov [10] to describe deformations of Lie algebras obeying a generalized Jacobi identity twisted by a homomorphism; the name. We describe their nuclei (Proposition 4.14), center (Corollary 4.16), and set of derivations (Proposition 4.18 and Proposition 4.19), and show that every nonzero endomorphism is injective (Proposition 4.21), but that there are nonzero endomorphisms which are not surjective (Proposition 4.22). We show that the hom-associative Weyl algebras are a multi-parameter formal hom-associative deformation of the first associative Weyl algebra (Proposition 5.2), inducing a multi-parameter formal hom-Lie deformation of the corresponding Lie algebra, when using the commutator as bracket (Proposition 5.4)

Preliminaries

Seven little lemmas on Yau twisted algebras

Multi-parameter formal deformations