Abstract
Here we explain some results about the polynomiality of the Poisson semicentre for parabolic subalgebras in a complex simple Lie algebra in the particular case of maximal parabolic subalgebras in a simple Lie algebra of type B.
I would like to dedicate this paper to Anthony Joseph for his 75th birthday, thanks to whom I discovered the world of quantum groups and then of (classical) enveloping algebras and with whom I worked a long time on this interesting subject of polynomiality of the Poisson semicentre associated to parabolic subalgebras.
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References
Borho, W., and P. Gabriel, and R. Rentschler, “Primideale in Einhüllenden auflösbarer Lie-Algebren (Beschreibung durch Bahnenräume)”, Lecture Notes in Math. 357, Springer-Verlag, Berlin, 1973.
Bourbaki, N., “Groupes et algèbres de Lie, Chapitres IV-VI”, Hermann, Paris, 1968.
Dixmier, J., Sur le centre de l’algèbre enveloppante d’une algèbre de Lie, C.R. Acad. Sc. Paris 265 (1967), 408–410.
Dixmier, J., “Algèbres enveloppantes”, Editions Jacques Gabay, les grands classiques Gauthier-Villars, Paris/Bruxelles/Montréal, 1974.
Fauquant-Millet, F., and A. Joseph, Sur les semi-invariants d’une sous-algèbre parabolique d’une algèbre enveloppante quantifiée, Transformation Groups 6 (2001), no. 2, 125–142.
Fauquant-Millet, F., and 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. (4) 38 (2005), no. 2, 155–191.
Fauquant-Millet, F., and A. Joseph, Adapted pairs and Weierstrass sections, http://arxiv.org/abs/1503.02523.
Fauquant-Millet, F., and P. Lamprou, Slices for maximal parabolic subalgebras of a semisimple Lie algebra, Transformation Groups 22, No. 4 (2017), 911–932.
Fauquant-Millet, F., and P. Lamprou, Polynomiality for the Poisson Centre of Truncated Maximal Parabolic Subalgebras, Journal of Lie Theory 28, No. 4 (2018), 1063–1094.
Joseph, A., A preparation theorem for the prime spectrum of a semisimple Lie algebra, J. Algebra 48 (1977), 241–289.
Joseph, A., On semi - invariants and index for biparabolic (seaweed) algebras I, J. Algebra 305 (2006), no. 1, 487–515.
Joseph, A., On semi - invariants and index for biparabolic (seaweed) algebras II, J. Algebra 312 (2007), no. 1, 158–193.
Joseph, A., Slices for biparabolic coadjoint actions in type A, J. Algebra 319 (2008), no. 12, 5060–5100.
Joseph, A., Compatible adapted pairs and a common slice theorem for some centralizers, Transform. Groups 13 (2008), no. 3–4, 637–669.
Joseph, A., An algebraic slice in the coadjoint space of the Borel and the Coxeter element, Advances in Maths 227 (2011), 522–585.
Joseph, A., and D. Shafrir, Polynomiality of invariants, unimodularity and adapted pairs, Transform. Groups 15 (2010), no. 4, 851–882.
Joseph, A., and P. Lamprou, Maximal Poisson commutative subalgebras for truncated parabolic subalgebras of maximal index in \(\mathfrak {sl}_n\), Transform. Groups 12 (2007), no. 3, 549–571.
Kostant, B., Lie group representations on polynomial rings, Amer. J. Math. 85 (1963), 327–404.
Panyushev, D., and A. Premet, and O. Yakimova, On symmetric invariants of centralisers in reductive Lie algebras, J. Algebra 313 (2007), no. 1, 343–391.
Acknowledgements
I would like to mention that this work is joint with Polyxeni Lamprou. More general results about polynomiality of the Poisson semicentre associated to a maximal parabolic subalgebra (not only in type B) are established in [9]. This work is the continuation of [8] where we also constructed adapted pairs in truncated maximal parabolic subalgebras \(\mathfrak a\), but in the case when we already knew the polynomiality of \(Y(\mathfrak a)\).
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Fauquant-Millet, F. (2019). About Polynomiality of the Poisson Semicentre for Parabolic Subalgebras. In: Gorelik, M., Hinich, V., Melnikov, A. (eds) Representations and Nilpotent Orbits of Lie Algebraic Systems. Progress in Mathematics, vol 330. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-23531-4_4
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DOI: https://doi.org/10.1007/978-3-030-23531-4_4
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