# Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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## Hochschild homology in a braided tensor categoryHTML articles powered by AMS MathViewer

by John C. Baez
Trans. Amer. Math. Soc. 344 (1994), 885-906 Request permission

## 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 ${H^R}(A)$.
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