Casimir Operators and Nilpotent Radicals

Ndogmo, J. C.

Canadian mathematical bulletin, Tome 55 (2012) no. 3, pp. 579-585

Résumé

It is shown that a Lie algebra having a nilpotent radical has a fundamental set of invariants consisting of Casimir operators. A different proof is given in the well known special case of an abelian radical. A result relating the number of invariants to the dimension of the Cartan subalgebra is also established.

Détail

DOI : 10.4153/CMB-2011-102-2

Mots-clés : 16W25, 17B45, 16S30, nilpotent radical, Casimir operators, algebraic Lie algebras, Cartan subalgebras, number of invariants

Bibliographie

[1] [1] Abellanas, L. and Alonso, L. Martinez, A general setting for Casimir invariants. J. Mathematical Phys. 16(1975), 1580–1584. Google Scholar | DOI

[2] [2] Bernat, P., Sur le corps des quotients de l’algèbre enveloppante d’une algèbre de Lie. C. R. Acad. Sci. Paris. 254(1962), 1712–1714. Google Scholar

[3] [3] Boyko, V., Patera, J., and Popovych, R., Invariants of triangular Lie algebras. J. Phys. A 40(2007), no. 27 7557–7572. Google Scholar | DOI

[4] [4] Boyko, V., Patera, J., and Popovych, R., Invariants of triangular Lie algebras with one nil-independent diagonal element. J. Phys. A 40(2007), no. 32, 9783–9792. Google Scholar | DOI

[5] [5] Campoamor-Stursberg, R., Some remarks concerning the invariants of rank one solvable real Lie algebras. Algebra Colloq. 12(2005), no. 3, 497–518. Google Scholar

[6] [6] Chevalley, C., Théorie des groupes de Lie. III. Hermann, Paris, 1955. Google Scholar

[7] [7] Dixmier, J., Sur l’algébre enveloppante d’une algébre de Lie nilpotente. Arch. Math. 10(1959), 321–326. Google Scholar | DOI

[8] [8] Dixmier, J., Sur les représentations unitaires des groupes de Lie nilpotents II, Bull. Soc. Math. France 85 (1957), 325–388. Google Scholar

[9] [9] Dixmier, J., Algèbres enveloppantes. Gauthier-Villars, Paris, 1974. Google Scholar

[10] [10] Humphreys, J. E., Introduction to Lie algebras and Representation Theory. Graduate Texts in Mathematics 9. Springer-Verlag, New York 1972. Google Scholar

[11] [11] Jacobson, N., Lie Algebras. Interscience Tracts in Pure and Applied Mathematics 10. John Wiley & Sons, New York, 1962. Google Scholar

[12] [12] Letellier, E. D., Deligne-Lusztig induction for invariant functions on finite Lie algebras of Chevalley's type. Tokyo J. Math. 28(2005), no. 1, 265–282. Google Scholar | DOI

[13] [13] Ndogmo, J. C., Invariants of a semi-direct sum of Lie algebras. J. Phys. A 37(2004), no. 21, 5635–5647. Google Scholar | DOI

[14] [14] Ndogmo, J. C., Properties of the invariants of solvable Lie algebras. Canad. Math. Bull. 43(2000), no. 4, 459–471. Google Scholar | DOI

[15] [15] Olver, P. J., Applications of Lie Groups to Differential Equations. Second edition. Graduate Texts in Mathematics 107. Springer-Verlag, New York, 1993. Google Scholar

[16] [16] Pecina-Cruz, J. N., An algorithm to calculate the invariants of any Lie algebra. J. Math. Phys. 35(1994), no. 6, 3146–3162. Google Scholar | DOI

[17] [17] Perroud, M., The fundamental invariants of inhomogeneous classical groups. J. Math. Phys 24(1983), no. 6, 1381–1391. Google Scholar | DOI

[18] [18] Racah, G., Sulla caratterizzazione delle rappresentazioni irriducibli dei gruppi semisemplici di Lie. Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat. 8(1950), 108–112. Google Scholar

[19] [19] Rentschler, R., Sur le centre du corps enveloppant d’une algèbre de Lie résoluble. C. R. Acad. Sci. Paris Sér. A-B 276(1973), A21–A24. Google Scholar

[20] [20] Snobl, L. and Winternitz, P., A class of solvable Lie algebras and their Casimir invariants. J. Phys. A 38(2005), no. 12, 2687–2700. Google Scholar | DOI

[21] [21] Snobl, L. and Winternitz, P., All solvable extensions of a class of nilpotent Lie algebras of dimension n and degree of nilpotency n – 1. J. Phys. A 42(2009), no. 10, 105201. http://dx.doi.org/10.1088/1751–8113/42/10/105201 Google Scholar