Article ID Journal Published Year Pages File Type
4589116 Journal of Algebra 2006 29 Pages PDF
Abstract

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 number is combinatorially more subtle than index q. As a by-product we show that Sq′(t) is always polynomial. The present methods are an adaption of those used in the study of similar questions for parabolic subalgebras in [F. Fauquant-Millet, A. Joseph, Sur les semi-invariants d'une sous-algèbre parabolique d'une algèbre enveloppante quantifiée, Transform. Groups 6 (2001) 125–142; F. Fauquant-Millet, A. Joseph, Semi-centre de l'algèbre enveloppante d'une sous-algèbre parabolique d'une algèbre de Lie semi-simple, Ann. Sci. École Norm. Sup. 38 (2005) 155–191; F. Fauquant-Millet, A. Joseph, La somme des faux degrés—un mystère en théorie des invariants, preprint, 2005]. Many problems remain open.

Related Topics
Physical Sciences and Engineering Mathematics Algebra and Number Theory