Transactions of the American Mathematical Society

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



Hochschild homology in a braided tensor category

Author: John C. Baez
Journal: Trans. Amer. Math. Soc. 344 (1994), 885-906
MSC: Primary 16W99; Secondary 16E40, 18G99
MathSciNet review: 1240942
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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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Article copyright: © Copyright 1994 American Mathematical Society