Involutive A-infinity algebras and dihedral cohomology

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.

22 pages, some minor typos corrected

References in corpus (4)

• Homotopy algebras and noncommutative geometry
• A dual point of view on the ribbon graph decomposition of moduli space
• Deformation Theory (lecture notes)
• Moduli spaces of Klein surfaces and related operads

Cited by in corpus (7)

• Quantisation of derived Lagrangians
• Quantisation of derived Poisson structures
• Reflexive homology
• Dihedral and reflexive modules with $\infty$-simplicial faces and dihedral and reflexive homology of involutive $A_{\infty }$-algebras over unital commutative rings
• Cohomology and deformations of oriented dialgebras
• Reflexive homology and involutive Hochschild homology as equivariant Loday constructions
• The homotopy invariance of dihedral homology of involutive $A_{\infty }$-algebras over rings