Deformation Cohomology for Braided Commutativity
Masahico Saito, Emanuele Zappala
TL;DR
The paper develops a braided-commutativity deformation theory for braided algebras by extending Yang–Baxter Hochschild cohomology to BC constraints. It proves that the BC second cohomology $H^2_{BC}(V,V)$ classifies braided-commutative infinitesimal deformations and that obstructions to higher-order BC deformations lie in $H^3_{BC}(V,V)$. It connects Hopf-algebra cocycles to BC cohomology via the adjoint $R$-matrix, producing nontrivial BC examples, and outlines a path toward higher dimensions through a multicomplex framework and a diagrammatic calculus. Together, these results provide a structured approach to understanding deformations under braided commutativity and suggest practical methods for constructing and obstructing higher-order BC deformations in braided algebraic settings.
Abstract
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 higher order 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 higher order deformations. We consider braided commutativity for Hopf algebras in detail, and obtain some classes of nontrivial examples.