Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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



Extensions of Hopf Algebras and Lie Bialgebras

Author: Akira Masuoka
Journal: Trans. Amer. Math. Soc. 352 (2000), 3837-3879
MSC (2000): Primary 16W30; Secondary 17B37, 17B56
Published electronically: March 24, 2000
MathSciNet review: 1624190
Full-text PDF

Abstract | References | Similar Articles | Additional Information


Let $\mathfrak{f}$, $\mathfrak{g}$ be finite-dimensional Lie algebras over a field of characteristic zero. Regard $\mathfrak{f}$ and $\mathfrak{g} ^*$, the dual Lie coalgebra of $\mathfrak{g}$, as Lie bialgebras with zero cobracket and zero bracket, respectively. Suppose that a matched pair $(\mathfrak{f} , \mathfrak{g} ^*)$of Lie bialgebras is given, which has structure maps $\rightharpoonup , \rho$. Then it induces a matched pair $(U\mathfrak{f}, U\mathfrak{g}^{\circ},\rightharpoonup ', \rho ')$ of Hopf algebras, where $U\mathfrak{f}$ is the universal envelope of $\mathfrak{f}$ and $U\mathfrak{g}^{\circ}$ is the Hopf dual of $U\mathfrak{g}$. We show that the group $\mathrm{Opext} (U\mathfrak{f},U\mathfrak{g}^{\circ })$of cleft Hopf algebra extensions associated with $(U\mathfrak{f}, U\mathfrak{g} ^{\circ}, \rightharpoonup ', \rho ' )$ is naturally isomorphic to the group $\operatorname{Opext}(\mathfrak{f},\mathfrak{g} ^*)$of Lie bialgebra extensions associated with $(\mathfrak{f}, \mathfrak{g}^*, \rightharpoonup , \rho )$. An exact sequence involving either of these groups is obtained, which is a variation of the exact sequence due to G.I. Kac. If $\mathfrak{g} =[\mathfrak{g} , \mathfrak{g}]$, there follows a bijection between the set $\mathrm{Ext}(U\mathfrak{f} , U\mathfrak{g}^{\circ })$of all cleft Hopf algebra extensions of $U\mathfrak{f}$ by $U\mathfrak{g}^{\circ }$ and the set $\mathrm{Ext}(\mathfrak{f}, \mathfrak{g}^*)$ of all Lie bialgebra extensions of $\mathfrak{f}$ by $\mathfrak{g} ^*$.

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

  • [A] N. Andruskiewitsch, Note on extensions of Hopf algebras (with an appendix by N. Andruskiewitsch and H.-J. Schneider), Can. J. Math. 48(1996), 3-42. MR 97c:16046
  • [BCM] R.J. Blattner, M. Cohen and S. Montgomery, Crossed products and inner actions of Hopf algebras, Trans. Amer. Math. Soc. 298(1986), 671-711. MR 87k:16012
  • [B] N. Bourbaki, ``Lie Groups and Lie Algebras", Part I, Hermann, Paris, 1975. MR 89k:17001 (reprint)
  • [B1] N. Bourbaki, ``General Topology", Part I, Hermann, Paris, 1966. MR 34:5044a
  • [CE] H, Cartan and S. Eilenberg, ``Homological Algebra", Princeton Univ. Press, Princeton, 1956. MR 17:1040e
  • [Doi] Y. Doi, Algebras with total integrals, Comm. Algebra 13(1985), 2137-2159. MR 87c:16013
  • [Don] S. Donkin, On the Hopf algebra dual of an enveloping algebra, Math. Proc. Cambridge Phil. Soc. 91(1982), 215-224. MR 83h:16014
  • [D] V. G. Drinfeld, Quantum groups, Proc. ICM-86, Berkeley, 1987, 798-820. MR 89f:17017
  • [HS] P. Hilton and U. Stammbach, ``A Course in Homological Algebra", Graduate Texts in Math., Vol.4, Springer-Verlag, Berlin, 1971. MR 49:10751
  • [H] G. Hochschild, Algebraic Lie algebras and representative functions, Illinois J. Math. 3(1959), 499-523. MR 23:A3207
  • [H1] G. Hochschild, Algebraic Lie algebras and representative functions, supplements, Illinois J. Math. 4(1960), 609-618. MR 22:A3208
  • [H2] G. Hochschild, Algebraic groups and Hopf algebras, Illinois J. Math. 14(1970), 52-65. MR 41:1742
  • [Hf] I. Hofstetter, Extensions of Hopf algebras and their cohomological description, J. Algebra 164(1994), 264-298. MR 95e:16035
  • [Kac] G. I. Kac, Extensions of groups to ring groups, Math. USSR Sbornik 5(1968), 451-474. MR 37:4639
  • [Kas] C. Kassel, ``Quantum Groups", Graduate Texts in Math., Vol.155, Springer-Verlag, New York, 1995. MR 96e:17041
  • [K] J. Koszul, Sur les modules de représentations des algèbres de Lie resolubles, Amer. Math. J. 76(1954), 535-554. MR 15:928e
  • [L] T. Levasseur, L'enveloppe injective du module trivial sur une algèbre de Lie resoluble, Bull. Sc. math., $2^{\text{e}}$ série 110(1986), 49-61. MR 88c:17015
  • [Mj] S. Majid, ``Foundations of Quantum Group Theory", Cambridge Univ. Press, Cambridge, 1995. MR 97g:17016
  • [M] A. Masuoka, Calculations of some groups of Hopf algebra extensions, J. Algebra 191(1997), 568-588; Corrigendum, J. Algebra 197(1997), 656. MR 98i:16042; MR 99f:16042
  • [MD] A. Masuoka and Y. Doi, Generalization of cleft comodule algebras, Comm. Algebra 20(1992), 3703-3721. MR 93j:16030
  • [Mi] W. Michaelis, Lie coalgebras, Adv. in Math. 38(1980), 1-54. MR 82g:17016
  • [Mo] S. Montgomery, ``Hopf Algebras and Their Actions on Rings", CBMS Regional Conference Series in Math., Vol.82, Amer. Math. Soc., Providence, 1993. MR 94i:16019
  • [Mo1] S. Montgomery, Crossed products of Hopf algebras and enveloping algebras, in: Perspective in Ring Theory, Kluwer, 1988, 253-268. MR 91i:16056
  • [R] J. J. Rotman, ``Introduction to Homological Algebra", Academic Press, New York, 1979. MR 80k:18001
  • [S] W. M. Singer, Extension theory for connected Hopf algebras, J. Algebra 21(1972), 1-16. MR 47:8597
  • [Sw] M. E. Sweedler, Cohomology of algebras over Hopf algebras, Trans. Amer. Math. Soc. 127(1968), 205-239. MR 37:283
  • [T] M. Takeuchi, Matched pairs of groups and bismash products of Hopf algebras, Comm. Algebra 9(1981), 841-882. MR 83f:16013
  • [T1] M. Takeuchi, Topological coalgebras, J. Algebra 87(1985), 505-539. MR 87e:16028
  • [Wa] W. C. Waterhouse, ``Introduction to Affine Group Schemes", Graduate Texts in Math., Vol.66, Springer-Verlag, New York, 1979. MR 82e:14003
  • [We] C. A. Weibel, ``Introduction to Homological Algebra", Cambridge Univ. Press, Cambridge, 1994. MR 95f:18001

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 16W30, 17B37, 17B56

Retrieve articles in all journals with MSC (2000): 16W30, 17B37, 17B56

Additional Information

Akira Masuoka
Affiliation: Mathematisches Institut der Universität München, Theresienstr. 39, D-80333 München, Germany; On leave of absence from: Department of Mathematics, Shimane University, Matsue, Shimane 690, Japan
Address at time of publication: Institute of Mathematics, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan

Keywords: Extension, Hopf algebra, Lie bialgebra, Lie algebra cohomology, continuous modules
Received by editor(s): May 23, 1997
Received by editor(s) in revised form: April 10, 1998
Published electronically: March 24, 2000
Additional Notes: This work was done at the Forschungsstipendiat der Alexander von Humboldt-Stiftung. The revision was done during a visit to the FaMAF, University of Córdoba. Their hospitality is gratefully acknowledged.
Dedicated: Dedicated to Professor Bodo Pareigis on his sixtieth birthday
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society