Bialgebra structures on a real semisimple Lie algebra
type: Item
instance of: scholarly article
title: Bialgebra structures on a real semisimple Lie algebra
zbMATH Open document ID: 0849.17009
author: Véronique Chloup
published in: Bulletin of the Belgian Mathematical Society - Simon Stevin
publication date: 1996-11-10T00:00:00Z
full work available at URL: https://eudml.org/doc/227616
review text: The Manin algebra \({\mathcal L}\) of the Lie bialgebra structure of a Lie algebra \(L\) with a 1-cocycle \(\psi: L\to \Lambda^2 L\) is defined in the following way: \({\mathcal L}\) has a nondegenerate invariant bilinear symmetric form \(\langle \langle\;,\;\rangle \rangle\); \({\mathcal L}\) is a direct sum (as vector spaces) of subalgebras \(L_1\) and \(L_2\); \(L_1\) and \(L_2\) are isotropic for \(\langle \langle\;,\;\rangle \rangle\). Theorem. Let \(L\) be a semisimple real Lie algebra; \(L= \bigoplus_{1\leq k\leq p} L_k\), where \(L_k\) are simple ideals of \(L\). Then up to isomorphism the Manin algebra \({\mathcal L}\) associated to \(L\) is of the form \({\mathcal L}= \bigoplus_{1\leq k\leq p} {\mathcal L}_k\), where \({\mathcal L}_k\) is one of the following: \({\mathcal L}_k= \text{Lie} (T^* (I_k))\), where \(I_k\) is the Lie group associated to the Lie algebra \(L_k\); \({\mathcal L}_k= (L_k )^C\); \({\mathcal L}_k= L_k \oplus L_k\). The bialgebra structures on a real semisimple Lie algebra for the nonzero standard modified Yang-Baxter equation are determined.
reviewed by: A. Dzhumadil'daev
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 16W30
zbMATH DE Number: 817246
zbMATH Keywords: Lie bialgebras
zbMATH Keywords: Manin triples
zbMATH Keywords: 1-cocycles
zbMATH Keywords: semisimple Lie algebras
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 817246