A remarkable contraction of semisimple Lie algebras
[Une contraction remarquable pour les algèbres de Lie semi-simples]
Panyushev, Dmitri I.1 ; Yakimova, Oksana S.2
1 Institute for Information Transmission Problems of the R.A.S. B. Karetnyi per. 19 Moscow 127994 (Russia)
2 Friedrich-Schiller-Universität Jena Mathematisches Institut Jena D-07737 (Deutschland)
Annales de l'Institut Fourier, Tome 62 (2012) no. 6, pp. 2053-2068.

E. Feigin a introduit la contraction 𝔮 d’une algèbre de Lie semi-simple 𝔤 dans arXiv :1007.0646 et arXiv :1101.1898. Nous démontrons que ces algèbres de Lie non-réductives conservent quelque unes des propriétés de 𝔤. En particulier, les algèbres des invariants des représentations adjointe et respectivement coadjointe de 𝔮 sont libres, et l’algèbre enveloppante de 𝔮 est un module libre sur son centre.

Recently, E.Feigin introduced a very interesting contraction 𝔮 of a semisimple Lie algebra 𝔤 (see arXiv:1007.0646 and arXiv:1101.1898). We prove that these non-reductive Lie algebras retain good invariant-theoretic properties of 𝔤. For instance, the algebras of invariants of both adjoint and coadjoint representations of 𝔮 are free, and also the enveloping algebra of 𝔮 is a free module over its centre.

DOI : 10.5802/aif.2742
Classification : 13A50, 14L30, 17B40, 22E46
Keywords: Inönü-Wigner contraction, coadjoint representation, algebra of invariants, orbit
Mot clés : contraction de Inönü-Wigner, représentation coadjointe, algebre des invariants, orbite
Panyushev, Dmitri I. 1 ; Yakimova, Oksana S. 2

1 Institute for Information Transmission Problems of the R.A.S. B. Karetnyi per. 19 Moscow 127994 (Russia)
2 Friedrich-Schiller-Universität Jena Mathematisches Institut Jena D-07737 (Deutschland) 
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Panyushev, Dmitri I.; Yakimova, Oksana S. A remarkable contraction of semisimple Lie algebras. Annales de l'Institut Fourier, Tome 62 (2012) no. 6, pp. 2053-2068. doi : 10.5802/aif.2742. http://www.numdam.org/articles/10.5802/aif.2742/

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