This paper is devoted to Hochschild homology of so-called \(r\)-algebras, i.e., algebras which are equipped with a Yang-Baxter operator \(R : A \otimes A \to A \otimes A\). The main source of examples of such algebras are algebras in braided tensor categories, and in fact any \(r\)-algebra can be obtained in that way.
The author first reviews the basic notions and results from the theory of braided tensor categories and algebras in these categories. In particular, he constructs the opposite algebra to such an algebra and the tensor product of two such algebras. Having these results at hand, one can form for an algebra \(A\) in a braided tensor category the algebra \(A^ e : = A \otimes A^{op}\), where \(A^{op}\) is the opposite algebra (in the braided sense) and the tensor product is the braided one. The original algebra \(A\) is canonically a left module over \(A^ e\), and in case that it is weakly \(r\)-commutative, i.e., \(\mu \circ R \circ R = \mu\), where \(\mu\) denotes the multiplication in \(A\), it is also a right module over \(A^ e\). Next, the author defines a flat resolution of \(A\) analogous to the standard acyclic Hochschild complex. If \(A\) is weakly \(r\)-commutative, he uses this to construct an analogue of the standard Hochschild complex, thus arriving at a definition of Hochschild homology.
Then he constructs the braided analog of the shuffle product and discusses relations between braided Hochschild homology and braided differential forms.