Co-Poisson Hopf structure

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last updates: 10 April 2011

Basic properties

[D1 Thm. 1] Let 𝔤 be a Lie algebra and let δ:𝔘𝔤𝔘𝔤𝔘𝔤 be a map that turns 𝔘𝔤 into a co-Poisson Hpf algebra with the standard multiplication and comultiplication. Then δ𝔤𝔤𝔤 , and the map φ:𝔤𝔤𝔤 indused by δ turns 𝔤 into a Lie bialgebra. Conversely, if 𝔤,φ is a Lie bialgebra, then there is a uniqe map δ:𝔘𝔤𝔘𝔤𝔘𝔤 which is an extension of φ and which turns 𝔘𝔤 into a co-Poisson Hopf algebra.



[Bou] N. Bourbaki, Lie groups and Lie algebras, Chapters I-III, Springer-Verlag, 1989.

[D] V.G. Drinfeld, Quantum Groups, Vol. 1 of Proccedings of the International Congress of Mathematicians (Berkeley, Calif., 1986). Amer. Math. Soc., Providence, RI, 1987, pp. 198–820. MR0934283

The theorem that a Lie bialgebra structure determinse a co-Poisson Hopf algebra structure on its enveloping algebra is due to Drinfel'd and appears as Theorem 1 in the following article.

[D1] V.G. Drinfel'd, Hopf algebras and the quantum Yang-Baxter equation, Soviet Math. Dokl. 32 (1985), 254–258. MR0802128

[DHL] H.-D. Doebner, Hennig, J. D. and W. Lücke, Mathematical guide to quantum groups, Quantum groups (Clausthal, 1989), Lecture Notes in Phys., 370, Springer, Berlin, 1990, pp. 29–63. MR1201823

The following book is a standard introductory book on Lie algebras; a book which remains a standard reference. This book has been released for a very reasonable price by Dover publishers.

[J] N. Jacobson, Lie algebras, Interscience Publishers, New York, 1962.

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