alexa On semi-invariants and index for biparabolic (seaweed) algebras, I


Journal of Generalized Lie Theory and Applications

Author(s): Anthony Joseph

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Biparabolic subalgebras of semisimple Lie algebras were introduced by V. Dergachev and A. Kirillov [V. Dergachev, A. Kirillov, Index of Lie algebras of seaweed type, J. Lie Theory 10 (2000) 331–343] under the name of Lie algebras of seaweed type. Let q be such an algebra, q′ its derived algebra, t its nilradical and S(q) the symmetric algebra over q. Now q is algebraic, so by a result of Chevalley–Dixmier [J. Dixmier, Sur les représentations unitaires des groupes de Lie nilpotents. II, Bull. Soc. Math. France 85 (1957) 325–388], index q=trdeg(FractSq(q)). Here we give a combinatorial formula for index q and use it to prove a conjecture of P. Tauvel and R.W.T. Yu [P. Tauvel R.W.T. Yu, Sur l'indice de certaines algèbres de Lie, Ann. Inst. Fourier (Grenoble) 54 (2004) 1793–1810]. We also compute the Gelfand–Kirillov dimension of Sq′(q), an algebra we conjecture to be polynomial.

This article was published in Journal of Algebra and referenced in Journal of Generalized Lie Theory and Applications

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