A remarkable contraction of semisimple Lie algebras

Panyushev, Dmitri I.; Yakimova, Oksana S.

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.

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

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