B-module algebras and their deformations

Let $A$ be an associative $k$-algebra and $B$ a bialgebra. A $B$-module algebra structure on $A$ is an action of $B$ on $A$ such that the comultiplication of $B$ is compatible with the multiplication of $A$, i.e.\ $b\circ {\mu }_a={\mu }_a\circ {\mathrm{\Delta }}_B(b)$. We discuss formal deformations of such structures, i.e.\ their behavior under the extension $k\to k\text{ha}$. For $B^{\prime }=B\text{ha}$ an element $P\in B^{\prime }\otimes B^{\prime }$ defines a universal deformation formula ${\mathrm{\Delta }}_h=P^{-1}\circ \mathrm{\Delta }\circ P$ for the comultiplication on $B^{\prime }$ and ${\mu }_h=\mu \circ P$ for the multiplication on $A^{\prime }=A\text{ha}$ if it is a solution of the compatibility condition ${\mathrm{\Delta }}_{1}P\cdot P_{12}={\mathrm{\Delta }}_{2}P\cdot P_{23}$. For $B=\mathcal{D}(H)$, the quantum double of the Hopf algebra $H$, the Quantum Yang-Baxter-Equation is a special case of this equation. Many special algebra deformations (e.g., quantizations) considered in the literature can be obtained this way, i.e.\ there is a (often hidden) $B$-module algebra structure and a universal deformation formula $P$ for them. The compatibility condition has the form of a Maurer-Cartan equation. It turns out that ${\mu }_h$ can be extended to a trivial deformation of the smash product $A\mathrm{\#}{B}^{\prime }$. Most of the equations arize as certain commutativity conditions of diagrams, that are best formulated in the language of categories. It is well known that the basic facts about quantum group theory or topological quantum field theory (TQFT) can be formulated in the language of monoidal categories. In this framework we will more generally discuss deformations of the structure of categories.