Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
Malcev algebras; universal enveloping algebras; universal alternative envelopes; differential operators; Bol algebras; analytic loops
Malcev algebras; universal enveloping algebras; universal alternative envelopes; differential operators; Bol algebras; analytic loops
Summary:
We first discuss the construction by Pérez-Izquierdo and Shestakov of universal nonassociative enveloping algebras of Malcev algebras. We then describe recent results on explicit structure constants for the universal enveloping algebras (both nonassociative and alternative) of the 4-dimensional solvable Malcev algebra and the 5-dimensional nilpotent Malcev algebra. We include a proof (due to Shestakov) that the universal alternative enveloping algebra of the real 7-dimensional simple Malcev algebra is isomorphic to the 8-dimensional division algebra of real octonions. We conclude with some brief remarks on tangent algebras of analytic Bol loops and monoassociative loops.
References:
[1] Akivis M.A., Goldberg V.V.: Local algebras of a differentiable quasigroup. Bull. Amer. Math. Soc. 43 (2006), 207–226. DOI 10.1090/S0273-0979-06-01094-9 | MR 2216110
[2] Akivis M.A., Shelekhov A.M.: Fourth-order alternators of a local analytic loop and three-webs of multidimensional surfaces. Soviet Math. 33 (1989), 13–18. MR 1008142
[3] Bremner M.R., Hentzel I.R., Peresi L.A., Usefi H.: Universal enveloping algebras of the four-dimensional Malcev algebra. Contemp. Math. 483 (2009), 73–89. DOI 10.1090/conm/483/09436 | MR 2497952
[4] Bremner M.R., Usefi H.: Enveloping algebras of the nilpotent Malcev algebra of dimension five. Algebr. Represent. Theory(to appear). MR 2660854
[5] Carlsson R.: Malcev-Moduln. J. Reine Angew. Math. 281 (1976), 199–210. MR 0393158 | Zbl 0326.17010
[6] Elduque A.: On maximal subalgebras of central simple Malcev algebras. J. Algebra 103 (1986), 216–227. DOI 10.1016/0021-8693(86)90181-X | MR 0860701 | Zbl 0592.17014
[7] Elduque A., Shestakov I.P.: Irreducible non-Lie modules for Malcev superalgebras. J. Algebra 173 (1995), 622–637. DOI 10.1006/jabr.1995.1106 | MR 1327872 | Zbl 0824.17005
[8] Gavrilov A.V.: Malcev extensions. Southeast Asian Bull. Math.(to appear). MR 2657651
[9] Kuzmin E.N.: Malcev algebras and their representations. Algebra Logika 7 (1968), 48–69. MR 0252468
[10] Kuzmin E.N.: Malcev algebras of dimension five over a field of characteristic zero. Algebra Logika 9 (1971), 691–700. MR 0283033
[11] Kuzmin E.N., Shestakov I.P.: Nonassociative structures. pages 197–280 of Algebra VI, Encyclopaedia of Mathematical Sciences 57, Springer, Berlin, 1995. MR 1360006
[12] Malcev A.I.: Analytic loops. Mat. Sbornik N.S. 36/78 (1955), 569–576. MR 0069190
[13] Mikheev P.O.: On a problem of Chern-Akivis-Shelekhov on hexagonal three-webs. Aequationes Math. 51 (1996), 1–11. DOI 10.1007/BF01831136 | MR 1372779 | Zbl 0845.53011
[14] Mikheev P.O., Sabinin L.V.: Infinitesimal theory of local analytic loops. Soviet Math. Dokl. 36 (1988), 545–548. MR 0924255 | Zbl 0659.53018
[15] Morandi P.J., Pérez-Izquierdo J.M., Pumplün S.: On the tensor product of composition algebras. J. Algebra 243 (2001), 41–68. DOI 10.1006/jabr.2001.8773 | MR 1851653
[16] Pérez-Izquierdo J.M.: An envelope for Bol algebras. J. Algebra 284 (2005), 480–493. DOI 10.1016/j.jalgebra.2004.09.038 | MR 2114566
[17] Pérez-Izquierdo J.M.: Algebras, hyperalgebras, nonassociative bialgebras and loops. Advances Math. 208 (2007), 834–876. DOI 10.1016/j.aim.2006.04.001 | MR 2304338
[18] Pérez-Izquierdo J.M., Shestakov I.P.: An envelope for Malcev algebras. J. Algebra 272 (2004), 379–393. DOI 10.1016/S0021-8693(03)00389-2 | MR 2029038
[19] Sagle A.A.: Malcev algebras. Trans. Amer. Math. Soc. 101 (1961), 426–458. DOI 10.1090/S0002-9947-1961-0143791-X | MR 0143791 | Zbl 0136.02103
[20] Shelekhov A.M.: Classification of multidimensional three-webs according to closure conditions. J. Soviet Math. 55 (1991), 2140–2168. DOI 10.1007/BF01095908 | Zbl 0705.53014
[21] Shestakov I.P.: Speciality problem for Malcev algebras and Poisson Malcev algebras. pages 365–371 of Nonassociative Algebra and its Applications, Dekker, New York, 2000. MR 1755366 | Zbl 0971.17019
[22] Shestakov I.P.: Speciality and deformations of algebras. pages 345–356 of Algebra, de Gruyter, Berlin, 2000. MR 1754680 | Zbl 1027.17001
[23] Shestakov I.P., Umirbaev U.U.: Free Akivis algebras, primitive elements, and hyperalgebras. J. Algebra 250 (2002), 533–548. DOI 10.1006/jabr.2001.9123 | MR 1899864 | Zbl 0993.17002
[24] Shestakov I.P., Zhelyabin V.N.: The Chevalley and Kostant theorems for Malcev algebras. Algebra Logika 46 (2007), 560–584. MR 2378631
[25] Zhevlakov K.A., Slinko A.M., Shestakov I.P., Shirshov A.I.: Rings That Are Nearly Associative. Academic Press, New York, 1982. MR 0668355 | Zbl 0487.17001
Search
Partner of
