Involutive A-infinity algebras and dihedral cohomology

TL;DR

This paper extends cohomology theories to A-infinity algebras with involution, generalizing dihedral cohomology, and explores their deformation theory and potential generalizations to other homotopy algebras.

Contribution

It introduces and studies involutive A-infinity algebra cohomology, linking it to deformation theory and extending classical concepts to a broader algebraic context.

Findings

Defined involutive A-infinity algebra cohomology

Established deformation theory results for these algebras

Discussed generalizations to other homotopy algebra structures

Abstract

We define and study the cohomology theories associated to A-infinity algebras and cyclic A-infinity algebras equipped with an involution, generalising dihedral cohomology to the A-infinity context. Such algebras arise, for example, as unoriented versions of topological conformal field theories. It is well known that Hochschild cohomology and cyclic cohomology govern, in a precise sense, the deformation theory of A-infinity algebras and cyclic A-infinity algebras and we give analogous results for the deformation theory in the presence of an involution. We also briefly discuss generalisations of these constructions and results to homotopy algebras over Koszul operads, such as L-infinity algebras or C-infinity algebras equipped with an involution.