Yang-Baxter Hochschild Cohomology

Abstract: Braided algebras are associative algebras endowed with a Yang-Baxter operator that satisfies certain compatibility conditions involving the multiplication. Along with Hochschild cohomology of algebras, there is also a notion of Yang-Baxter cohomology, which is associated to any Yang-Baxter operator. In this article, we introduce and study a cohomology theory for braided algebras in dimensions 2 and 3, that unifies Hochschild and Yang-Baxter cohomology theories, and generalizes to all dimensions in characteristic $2$. We show that its second cohomology group classifies infinitesimal deformations of braided algebras. We provide infinite families of examples of braided algebras, including Hopf algebras, tensorized multiple conjugation quandles, and braided Frobenius algebras. Moreover, we derive the obstructions to higher deformations, which lie in the third cohomology group. Relations to Hopf algebra cohomology are also discussed.