Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-10-10256-1
Published electronically: January 22, 2010
MathSciNet review: 2596026
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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]$.


References [Enhancements On Off] (What's this?)

  • 1. Cartier, P.: Hyperalgèbres et groupes de Lie formels. Séminaire Sophus Lie, 2e année 1955/56, Faculté des Sciences de Paris.
  • 2. Connes, Alain; Moscovici, Henri: Cyclic cohomology and Hopf algebra symmetry. Conference Moshé Flato 1999 (Dijon). Lett. Math. Phys., 52 (2000), no. 1, 1-28. MR 1800488 (2002d:58009)
  • 3. Connes, Alain; Moscovici, Henri: Cyclic cohomology and Hopf algebras. Lett. Math. Phys., 48 (1999), no. 1, 97-108. MR 1718047 (2000j:16061)
  • 4. Connes, A.; Moscovici, H.: Hopf algebras, cyclic cohomology and the transverse index theorem. Comm. Math. Phys., 198 (1998), no. 1, 199-246. MR 1657389 (99m:58186)
  • 5. Drinfel'd, V.: Quantum Groups, Proc. Internat. Congress Mathematicians (Berkeley, 1986), Vol. 1, Amer. Math. Soc., Providence, RI, 1987, 798-820. MR 934283 (89f:17017)
  • 6. Hajac, Piotr M.; Khalkhali, Masoud; Rangipour, Bahram; Sommerhäuser, Yorck: Hopf-cyclic homology and cohomology with coefficients. C. R. Math. Acad. Sci. Paris, 338 (2004), no. 9, 667-672. MR 2065371 (2005b:19002)
  • 7. Hajac, Piotr M.; Khalkhali, Masoud; Rangipour, Bahram; Sommerhäuser, Yorck: Stable anti-Yetter-Drinfeld modules. C. R. Math. Acad. Sci. Paris, 338 (2004), no. 8, 587-590. MR 2056464 (2005a:16056)
  • 8. Loday, Jean-Louis: Cyclic Homology, Grundlehren der Mathematischen Wissenschaften, 301, Springer-Verlag, Berlin, 1998. MR 1600246 (98h:16014)
  • 9. Milnor, John W.; Moore, John C.: On the structure of Hopf algebras. Ann. of Math. (2), 81 (1965), 211-264. MR 0174052 (30:4259)
  • 10. Reshetikhin, N. Yu.; Turaev, V. G.: Ribbon graphs and their invariants derived from quantum groups. Comm. Math. Phys., 127 (1990), 1-26. MR 1036112 (91c:57016)
  • 11. Sweedler, M. E.: Hopf Algebras, Mathematics Lecture Note Series, 44, W. A. Benjamin, Inc., New York, 1969. MR 0252485 (40:5705)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 16W25, 16T05, 57T05

Retrieve articles in all journals with MSC (2010): 16W25, 16T05, 57T05


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
Email: abanerje@math.jhu.edu

DOI: https://doi.org/10.1090/S0002-9939-10-10256-1
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.

American Mathematical Society