Deformation Cohomology for Braided Commutativity

Braided algebras are algebraic structures consisting of an algebra endowed with a Yang-Baxter operator, satisfying some compatibility conditions. Yang-Baxter Hochschild cohomology was introduced by the authors to classify infinitesimal deformations of braided algebras, and determine obstructions to quadratic deformations. Several examples of braided algebras satisfy a weaker version of commutativity, which is called braided commutativity and involves the Yang-Baxter operator of the algebra. We extend the theory of Yang-Baxter Hochschild cohomology to study braided commutative deformations of braided algebras. The resulting cohomology theory classifies infinitesimal deformations of braided algebras that are braided commutative, and provides obstructions for braided commutative quadratic deformations. We consider braided commutativity for Hopf algebras in detail, and obtain some classes of nontrivial examples.