Please use this identifier to cite or link to this item: http://hdl.handle.net/1942/13842
Full metadata record
DC FieldValueLanguage
dc.contributor.authorOOMS, Alfons-
dc.date.accessioned2012-07-24T06:16:11Z-
dc.date.available2012-07-24T06:16:11Z-
dc.date.issued2012-
dc.identifier.citationJOURNAL OF ALGEBRA, 365, p. 83-113-
dc.identifier.issn0021-8693-
dc.identifier.urihttp://hdl.handle.net/1942/13842-
dc.description.abstractLet g be a finite dimensional Lie algebra over an algebraically closed field k of characteristic zero. We collect some general results on the Poisson center of S(g), including some simple criteria regarding its polynomiality, and also on certain Poisson commutative subalgebras of S(g). These facts are then used to finish our earlier work on this subject, i.e. to give an explicit description for the Poisson center of all indecomposable, nilpotent Lie algebras of dimension at most seven. Among other things, we also provide a polynomial, maximal Poisson commutative subalgebra of S(g), enjoying additional properties. As a by-product we show that a conjecture by Milovanov is valid in this situation. These results easily carry over to the enveloping algebra U(g).-
dc.language.isoen-
dc.rights2012 Elsevier Inc. All rights reserved-
dc.subject.otherPoisson center; Poisson commutative subalgebra; Nilpotent Lie algebra; Enveloping algebra-
dc.titleThe Poisson center and polynomial, maximal Poisson commutative subalgebras, especially for nilpotent Lie algebras of dimension at most seven-
dc.typeJournal Contribution-
dc.identifier.epage113-
dc.identifier.spage83-
dc.identifier.volume365-
local.bibliographicCitation.jcatA1-
dc.relation.references[AG] J.M. Ancochea-Bermudez, M. Goze, Classi cation des alg ebres de Lie nilpotentes complexes de dimension 7, Arch. Math. 52 (1989), 157{185. [AVE] E. Andreev, E. Vinberg, A.G. Elashvili, Orbits of greatest dimension in semisimple linear Lie groups, Functional Anal. Appl., v1 (1968), 257-261. [Bo1] A.V. Bolsinov, Commutative families of functions related to consistent Poisson brackets, Acta Appl. Math. 24 (1991), no 3, 253{274. [Bo2] A.V. Bolsinov, Complete commutative subalgebras in polynomial Poisson algebras: a proof of the Mishchenko-Fomenko conjecture (2008), paper available in PDF format, http://www-sta .lboro.ac.uk/ maab2/publications.html [BGR] W. Borho, P. Gabriel, R. Rentschler, Primideale in Einh ullenden au osbarer Lie-Algebren, Lecture Notes in Math., vol. 357, Springer-Verlag, Berlin, 1973. [Bou] N. Bourbaki, Groupes et alg ebres de Lie, chap. I (2nd ed.), Act. Sci. Ind. 1285, Hermann, Paris, 1971. [Ca] R. Carles, Weight systems for complex nilpotent Lie algebras and application to the varieties of Lie algebras, Pr epublication no 96, D epartement de Math ematiques, Universit e de Poitiers (1996). [Ce1] A. Cerezo, Les alg ebres de Lie nilpotentes, r eelles et complexes de dimension 6, Pr epublication 27, D epartement de Math ematiques, Universit e de Nice (1983), 34p. http://math.unice.fr/ frou/ACpublications.html [Ce2] A. Cerezo, On the rational invariants of a Lie algebra, Pr epublication 68, D epartement de Math ematiques, Universit e de Nice (1985), 54p. http://math.unice.fr/ frou/ACpublications.html [Ce3] A. Cerezo, Calcul des invariants alg ebriques et rationnels d'une matrice nilpotente, Pr epublication 35, D epartement de Math ematiques, Universit e de Poitiers (1988), 25p. http://math.unice.fr/ frou/ACpublications.html [Ce4] A. Cerezo, Sur les invariants alg ebriques du groupe engendr e par une matrice nilpotente, unpublished, handwritten manuscript, 83 p. http://math.unice.fr/ frou/ACinvariants/inv0.PDF (References and Introduction, 330 Kb) http://math.unice.fr/ frou/ACinvariants/inv1.PDF (Chapter I, 1291 Kb) http://math.unice.fr/ frou/ACinvariants/inv2.PDF (Chapter II, 2567 Kb) [DNO] L. Delvaux, E. Nauwelaerts, A.I. Ooms, On the semicenter of a universal enveloping algebra, J. Algebra 94 (1985), 324{346. [DNOW] L. Delvaux, E. Nauwelaerts, A.I. Ooms, P. Wauters, Primitive localization of an enveloping algebra, J. Algebra 130 (1990), 311{327. [D1] J. Dixmier, Sur les repr esentations unitaires des groupes de Lie nilpotents, II, Bull. Soc. Math. France 85 (1957), 325{388. [D2] J. Dixmier, Sur les repr esentations unitaires des groupes de Lie nilpotents, III, Canadian J. Math., 10, (1958), 321{348. [D3] J. Dixmier, Sur les repr esentations unitaires des groupes de Lie nilpotents, IV, Canadian J. Math., 11, (1959), 321{344. [D4] J. Dixmier, Sur les alg ebres enveloppantes de sl(n;C) et af(n;C), Bull. Sci. Math. 100 (1976), 57{95. [D5] J. Dixmier, Enveloping Algebras, Grad. Stud. Math., vol 11, Amer. Math. Soc., Providence, RI, 1996. [DDV] J. Dixmier, M. Du o, M. Vergne, Sur la repr esentation coadjointe d'une alg ebre de Lie, Compositio Math. 29 (1974), 309{323. [E] A. G. Elashvili, Frobenius Lie algebras, Funct. Anal. i Prilozhen 16 (1982) 94{95. [EO] A. G. Elashvili, A. I. Ooms, On commutative polarizations, J. Algebra 264 (2003), 129{154. [FJ1] F. Fauquant-Millet, A. Joseph, Semi-centre de l'alg ebre enveloppante d'une sous-alg ebre parabolique d'une alg ebre de Lie semi-simple, Ann. Sci. Ecole Norm. Sup. 38 (2005), 155{191. [FJ2] F. Fauquant-Millet, A. Joseph, La somme des faux degr es - un myst ere en th eorie des invariants, Adv. Math. 217 (2008), 1476{1520. [FS] G. Favre, L.J. Santharoubane, Symmetric, invariant, non-degenerate bilinear form on a Lie algebra, J. Algebra 105 (1987), 451{464. [GK] I.M. Gelfand, A.A. Kirillov, Sur les corps li es aux alg ebres enveloppantes des alg ebres de Lie, Inst. Hautes Etudes Sci. Publ. Math. 31 (1966) 5{19. [G] M-P. Gong, Classi cation of nilpotent Lie algebras of dimension 7, PhD Thesis, University of Waterloo, Ontario, Canada, 1998. [GoKh] M. Goze, Y. Khakimdjanov, Nilpotent Lie algebras, Mathematics and its applications, vol. 361, Kluwer, 1996. [GY] J. H. Grace, A. Young, The algebra of invariants, Cambridge University Press, 1903. [GPS] G.-Greuel,G. P ster, H. Sch onemann, SINGULAR 3.0, A Computer Algebra System for Polynomial Computations, Centre for Computer Algebra, University of Kaiserslautern, 2005, http://www.singular.uni-kl.de. [J1] A. Joseph, Proof of the Gelfand-Kirillov conjecture for solvable Lie algebras, Proc. Amer. Math. Soc. 45 (1974) 1-10. [J2] A. Joseph, A preparation theorem for the prime spectrum of a semisimple Lie algebra, J. Algebra 48 (1977) 241{289. [J3] A. Joseph, Parabolic actions in type A and their eigenslices, Transform. Groups, 12 (2007), no 3, 515{547. [J4] A. Joseph, Slices for biparabolic coadjoint actions in type A, J. Algebra 319 (2008), 5060{5100. [J5] A. Joseph, Compatible adapted pairs and a common slice theorem for some centralizers, Transform. Groups, 13 (2008), 637{669. [J6] A. Joseph, Invariants and slices for reductive and biparabolic coadjoint actions, Lecture Notes Weizmann, 20/2/2007, revised 7/1/2010, http://www.wisdom.weizmann.ac.il/ gorelik/agrt.htm. [JL] A. Joseph, P. Lamprou, Maximal Poisson Commutative subalgebras for truncated parabolic subalgebras of maximal index in sln, Transform. Groups, 12 (2007), 549{571. [JS] A. Joseph, D. Shafrir, Polynomiality of invariants, unimodularity and adapted pairs. Transform. Groups, 15 (2010), 851{882. [K] A.A. Korotkevich, Integrable Hamiltonian systems on low-dimensional Lie algebras, Sb. Math. 200, No 12, 1731{1766 (2009). [KPV] V.G. Kac, V.I. Popov, E.B. Vinberg, Sur les groupes lin eaires dont l'alg ebre des invariants est libre, C.R. Acad. Sc. Paris 283 (1976), 875-878. [Ma1] L. Magnin, Sur les alg ebres de Lie nilpotentes de dimension 7, J. Geom. Phys., 3 (1986), 119{144. [Ma2] L. Magnin, Adjoint and trivial cohomology tables for indecomposable nilpotent Lie algebras of dimension 7 over C, Electronic Book, Institut de Math ematique, Universit e de Bourgogne, Second Corrected Edition (2007), (810 pages + vi), http://www.u-bourgogne.fr/monge/l.magnin [Ma3] L. Magnin, Determination of 7-dimensional indecomposable nilpotent complex Lie algebras by adjoining a derivation to 6-dimensional Lie algebras, Algebr. Represent. Theory, 13 (2010), 723{753. [MF] A.S. Mishchenko, A.T. Fomenko, Euler equations on nite-dimensional Lie groups, Math. USSR-Izv. 12 (1978), 371{389. [MW] C.C. Moore, J.A. Wolf, Square integrable representations of nilpotent groups, Trans. Amer. Math. Soc. 185 (1973), 445{462. [M] V. Morozov, Classi cation of nilpotent Lie algebras of sixth order, Izv. Vyssh. Uchebn. Zaved. Mat., 4(5), (1958), 161{171. [N] Nghi^em Xu^an Hai, Sur certains sous-corps commutatifs du corps enveloppant d'une alg ebre de Lie r esoluble, Bull. Sci. Math. 96 (1972), 111{128. [O1] A.I. Ooms, On Lie algebras with primitive envelopes, supplements, Proc. Amer. Math. Soc. 58 (1976), 67{72. [O2] A.I. Ooms, On Frobenius Lie algebras, Comm. Algebra 8 (1980) 13{52. [O3] A.I. Ooms, On certain maximal sub elds in the quotient division ring of an enveloping algebra, J. Algebra, 230 (2000), 694{712. [O4] A. I. Ooms, The Frobenius semiradical of a Lie algebra, J. Algebra 273 (2004), 274{287. [O5] A. I. Ooms, Computing invariants and semi-invariants by means of Frobenius Lie algebras, J. Algebra 321 (2009), 1293{1312. [OV] A.I. Ooms, M. Van den Bergh, A degree inequality for Lie algebras with a regular Poisson semicenter, J. Algebra 323 (2010) 305{322, arXiv: math.RT /0805.1342v1, 2008. [P] D.I. Panyushev, On the coadjoint representation of Z2-contractions of reductive Lie algebras, Adv. Math. 213 (2007), no 1, 380-404. [PPY] D.I. Panyushev, A. Premet, O. S. Yakimova, On symmetric invariants of centralizers in reductive Lie algebras, J. Algebra 313 (2007), no 1, 343{391. [PY] D.I. Panyushev, O. S. Yakimova, The argument shift method and maximal commutative subalgebras of Poisson algebras, Math. Res. Lett.15 (2008), no 2, 239{249. arXiv:math.RT/0702583v1 (2007). [PSW] J. Patera, R. T. Sharp, P. Winternitz, H. Zassenhaus, Invariants of real low dimension Lie algebras, J. Math. Phys., 17 (1976), 986{994. [RV] R. Rentschler, M. Vergne, Sur le semi-centre du corps enveloppant d'une alg ebre de Lie, Ann. Sci. Ecole Norm. Sup. 6 (1973) 389{405. [R] M. Romdhani, Classi cation of real and complex nilpotent Lie algebras of dimension 7, Linear and Multilinear Algebra, 24 (1989) 167{189. [Sa] S. T. Sadetov, A proof of the Mishchenko-Fomenko conjecture (1981), Dokl. Akad. Nauk 397 (6) (2004), 751{754. [Sc] G.W. Schwarz, Representations of simple Lie groups with regular rings of invariants, Invent. Math. 49 (1978), 167-191. [Se] C. Seeley, 7-dimensional nilpotent Lie algebras, Trans. Amer. Math. Soc., 335 (1993) 479{496. [Sk] S. Skryabin, Invariants of nite group schemes, J. London Math. Soc. (2) 65 (2002), no. 2, 339{360. [T] A.A. Tarasov, The maximality of certain commutative subalgebras in Poisson algebras of a semisimple Lie algebra, Russian Math. Surveys 57 (2002), no. 5, 1013{1014. [TY] P. Tauvel, R. T. W. Yu, Lie Algebras and Algebraic Groups, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2005. [V] M. Vergne, La structure de Poisson sur l'alg ebre sym etrique d'une alg ebre de Lie nilpotente, Bull. Soc. Math. France, 100 (1972), 301{335. [Y] O. Yakimova, A counterexample to Premet's and Joseph's conjectures, Bull. London Math. Soc., 39 (2007) 749{754.-
local.type.refereedRefereed-
local.type.specifiedArticle-
dc.bibliographicCitation.oldjcatA1-
dc.identifier.doi10.1016/j.jalgebra.2012.04.029-
dc.identifier.isi000305776900006-
dc.identifier.urlhttp://arxiv.org/abs/1110.0363-
item.contributorOOMS, Alfons-
item.accessRightsOpen Access-
item.fullcitationOOMS, Alfons (2012) The Poisson center and polynomial, maximal Poisson commutative subalgebras, especially for nilpotent Lie algebras of dimension at most seven. In: JOURNAL OF ALGEBRA, 365, p. 83-113.-
item.fulltextWith Fulltext-
item.validationecoom 2013-
crisitem.journal.issn0021-8693-
crisitem.journal.eissn1090-266X-
Appears in Collections:Research publications
Files in This Item:
File Description SizeFormat 
PoissonCenterPolynomialMaximal_2.pdfMain article325.17 kBAdobe PDFView/Open
Show simple item record

Google ScholarTM

Check

Altmetric


Items in DSpace are protected by copyright, with all rights reserved, unless otherwise indicated.