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Transactions of the American Mathematical Society
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
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Abstract:

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 $\mathfrak{g} ^*$.


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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
Email: akira@math.tsukuba.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9947-00-02394-1
PII: S 0002-9947(00)02394-1
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