Deforming algebras with anti-involution via twisted associativity

TL;DR

This paper explores a novel deformation method for algebras with anti-involution, transforming them into Hom-associative algebras of type II, and develops a module theory and operadic framework for these structures.

Contribution

It introduces a deformation via anti-involution that yields Hom-associative algebras of type II and establishes a functorial link between module categories.

Findings

Twisting multiplication by anti-involution produces Hom-associative algebras of type II.

A faithful functor relates modules over involutive algebras to modules over Hom-associative algebras.

Operadic approach proposed for generalizing deformation studies.

Abstract

This contribution studies a specific deformation of algebras with anti-involution. Starting with the observation that twisting the multiplication of such an algebra by its anti-involution generates a Hom-associative algebra of type II, it formulates the adequate modules theory over these algebras, and shows that there is a faithful functor from the category of finite-dimensional left modules of algebras with involution to finite-dimensional right modules of Hom-associative algebras of type II. It ends with a discussion on a diagrammatic operadic approach to generalize the study of such deformations.