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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Hopf cyclic cohomology and biderivations

Author: Abhishek Banerjee
Journal: Proc. Amer. Math. Soc. 138 (2010), 1929-1939
MSC (2010): Primary 16W25, 16T05, 57T05
Published electronically: January 22, 2010
MathSciNet review: 2596026
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Abstract: Hopf cyclic cohomology $ HC^*_{(\delta,\sigma)}(\mathcal H)$ for a Hopf algebra $ \mathcal H$ with respect to a modular pair in involution $ (\delta,\sigma)$ was introduced by Connes and Moscovici. By a biderivation $ D$ on a Hopf algebra $ \mathcal H$ we shall mean a linear map that satisfies the axioms for both a derivation and a coderivation on $ \mathcal H$. Given a biderivation $ D$ on a Hopf algebra, we define, under certain conditions, a map $ L_D:HC^*_{(\delta,\sigma)}(\mathcal H)\longrightarrow HC^*_{(\delta,\sigma)}(\mathcal H)$. We give examples of such maps for the quantized universal enveloping algebra $ \mathcal U_h(\mathfrak{g})$ of a simple Lie algebra $ \mathfrak{g}$. When $ \mathcal H$ is irreducible, cocommutative and equipped with a character $ \delta$ such that $ (\delta,1)$ is a modular pair in involution, we define ``inner biderivations'' and use these to produce a left $ \mathcal H$-module structure on $ HC^*_{(\delta,1)}(\mathcal H)$. Finally, we show that every morphism $ L_D:HC^*_{(\delta,1)}(\mathcal H)\longrightarrow HC^*_{(\delta,1)}(\mathcal H)$ induced by a biderivation $ D$ on such a Hopf algebra $ \mathcal H$ can be realized as a morphism induced by an inner biderivation by embedding $ \mathcal H$ into a larger Hopf algebra $ \mathcal H[D]$.

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Additional Information

Abhishek Banerjee
Affiliation: Department of Mathematics, Johns Hopkins University, 3400 N. Charles Street, Baltimore, Maryland 21218
Address at time of publication: Department of Mathematics, Ohio State University, 231 W. 18th Avenue, Columbus, Ohio 43210

Keywords: Hopf cyclic cohomology, derivations, coderivations
Received by editor(s): April 9, 2009
Received by editor(s) in revised form: September 13, 2009
Published electronically: January 22, 2010
Communicated by: Varghese Mathai
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.