Hochschild homology in a braided tensor category

Abstract:

An r-algebra is an algebra A over k equipped with a Yang-Baxter operator $R:A \otimes A \to A \otimes A$ such that $R(1 \otimes a) = a \otimes 1$, $R(a \otimes 1) = 1 \otimes a$, and the quasitriangularity conditions $R(m \otimes I) = (I \otimes m)(R \otimes I)(I \otimes R)$ and $R(I \otimes m) = (m \otimes I)(I \otimes R)(R \otimes I)$ hold, where $m:A \otimes A \to A$ is the multiplication map and $I:A \to A$ is the identity. R-algebras arise naturally as algebra objects in a braided tensor category of k-modules (e.g., the category of representations of a quantum group). If $m = m{R^2}$, then A is both a left and right module over the braided tensor product ${A^e} = A\hat \otimes {A^{{\text {op}}}}$, where ${A^{{\text {op}}}}$ is simply A equipped with the "opposite" multiplication map ${m^{{\text {op}}}} = mR$. Moreover, there is an explicit chain complex computing the braided Hochschild homology ${H^R}(A) = \operatorname {Tor}^{{A^e}}(A,A)$. When $m = mR$ and ${R^2} = {\text {id}}_{A \otimes A}$, this chain complex admits a generalized shuffle product, and there is a homomorphism from the r-commutative differential forms ${\Omega _R}(A)$ to ${H^R}(A)$.