The Chevalley and Costant theorems for Mal'tsev algebras

V. N. Zhelyabin ; I. P. Shestakov

Algebra i logika, Tome 46 (2007) no. 5, pp. 560-584.

Résumé

Centers of universal envelopes for Mal'tsev algebras are explored. It is proved that the center of the universal envelope for a finite-dimensional semisimple Mal'tsev algebra over a field of characteristic 0 is a ring of polynomials in a finite number of variables equal to the dimension of its Cartan subalgebra, and that universal enveloping algebra is a free module over its center. Centers of universal enveloping algebras are computed for some Mal'tsev algebras of small dimensions.

Détail

Keywords: Lie algebra, Mal'tsev algebra, bialgebra, universal enveloping algebra, primitive elements, center of algebra, Chevalley theorem, Costant theorem.

Bibliographie

[1] Zh. Diksme, Universalnye obërtyvayuschie algebry, Mir, M., 1978 | MR

[2] A. I. Maltsev, “Analiticheskie lupy”, Matem. sb., 36(78):3 (1955), 569–576

[3] E. N. Kuzmin, I. P. Shestakov, “Neassotsiativnye struktury”, Algebra-6, Itogi nauki i tekhniki. Sovr. probl. matem. Fundam. napravl., 57, VINITI, M., 1990, 179–266 | MR

[4] A. A. Sagle, “Simple Malcev algebras over fields of characteristic zero”, Pac. J. Math., 12 (1962), 1057–1078 | MR | Zbl

[5] E .N. Kuzmin, “Algebry Maltseva i ikh predstavleniya”, Algebra i logika, 7:4 (1968), 48–69 | MR

[6] I. P. Shestakov, “Speciality problem for Malcev algebras and Poisson Malcev algebras”, Non-associative algebra and its applications, Proc. IV int. conf. non-associative algebra and its appl. (July 1998, São Paulo), Marcel Dekker, New York, 2000, 365–371 | MR | Zbl

[7] J. M. Pérez-Izquierdo, I. P. Shestakov, “An envelope for Malcev algebras”, J. Algebra, 272:1 (2004), 379–393 | DOI | MR | Zbl

[8] J. M. Pérez-Izquierdo, Algebras, hiperalgebras, nonassociative bialgebras and loops, Preprint, 2004 ; http//mathematik.ubik.ac.at/mathematik/jnordan | MR

[9] P. J. Moradni, J. M. Pérez-Izquierdo, S. Pumplüm, “On the tensor product of composition algebras”, J. Algebra, 243 (2001), 41–68 | DOI | MR

[10] I. P. Shestakov, U. U. Umirbaev, “Free Akivis algebras, primitive elements and hyperalgebras”, J. Algebra, 250:2 (2002), 533–548 | DOI | MR | Zbl

[11] K. A. Zhevlakov, A. M. Slinko, I. P. Shestakov, A. I. Shirshov, Koltsa, blizkie k assotsiativnym, Nauka, M., 1978 | MR | Zbl

[12] E. N. Kuzmin, “Struktura i predstavleniya konechnomernykh algebr Maltseva”, Issledovaniya po teorii kolets i algebr, Trudy In-ta matem. SO AN SSSR, 16, Nauka, Novosibirsk, 1989, 75–101 | MR

[13] J. M. Pérez-Izquierdo, “An envelope for Bol algebras”, J. Algebra, 284:2 (2005), 480–493 | DOI | MR | Zbl

[14] P. Miheev, L. Sabinin, “Quasigroups and differential geometry”, Quasigroups and loops: theory and applications, Sigma Ser. Pure Math., 8, Heldermann, Berlin, 1990, 357–430 | MR

[15] A. A. Sagle, “Malcev algebras”, Trans. Am. Math. Soc., 101:3 (1962), 426–458 | DOI | MR

[16] A. T. Gainov, “Binarno lievy algebry nizkikh rangov”, Algebra i logika, 2:4 (1963), 21–40 | MR | Zbl

[17] E. N. Kuzmin, “Algebry Maltseva razmernosti pyat nad polem kharakteristiki nul”, Algebra i logika, 9:6 (1970), 691–700 | MR

[18] A. Elduque, “On maximal subalgebras of central simple Malcev algebras”, J. Algebra, 103 (1986), 216–227 | DOI | MR | Zbl

[19] M. R. Bremner, H. Usefi, Structure constants for the universal enveloping algebra of the five-dimensional nilpotent non-Lie Malcev algebra, Preprint

[20] O. N. Smirnov, “Teorema Artina dlya kosoalternativnykh algebr”, Algebra i logika, 33:6 (1994), 708–711 | MR | Zbl