On the Cohomology of Cyclic Associative Algebras

Abstract

We introduce a cohomology theory for cyclic associative algebras, a subclass of shift associative algebras defined by the identity $(xy)z=x(yz)=y(zx)$. This cohomology, denoted $H_{\mathrm{cyc}}^{\bullet }(A,M)$, is a subtheory of Hochschild cohomology obtained by restricting to cochains that satisfy a cyclic compatibility condition derived from the defining identity. We prove that $H_{\mathrm{cyc}}^2(A,M)$ classifies cyclic associative extensions of $A$ by a cyclic bimodule $M$. The universal derivation and the module of differential forms ${\Omega }_{\mathbb{F}}^{\bullet }(A)$ are constructed, and $({\Omega }_{\mathbb{F}}^{\bullet }(A),d)$ is shown to be the universal cyclic differential graded algebra over $A$. For trivial coefficients, we establish natural inclusions $H{C}^n(A)\hookrightarrow H_{\mathrm{cyc}}^n(A,\mathbb{F})\hookrightarrow H{H}^n(A,\mathbb{F})$, placing our theory intermediate between Connes' cyclic cohomology and Hochschild cohomology.