Inner ideal structure of nearly Artinian Lie algebras

López, Antonio; García, Esther; Lozano, Miguel

doi:10.1090/S0002-9939-08-09607-X

Inner ideal structure of nearly Artinian Lie algebras
HTML articles powered by AMS MathViewer

by Antonio Fernández López, Esther García and Miguel Gómez Lozano PDF

Proc. Amer. Math. Soc. 137 (2009), 1-9 Request permission

Abstract:

In this paper we study the inner ideal structure of nondegenerate Lie algebras with essential socle, and characterize, in terms of the whole algebra, conditions under which the socle is Artinian.

References

Georgia Benkart, The Lie inner ideal structure of associative rings, J. Algebra 43 (1976), no. 2, 561–584. MR 435149, DOI 10.1016/0021-8693(76)90127-7
Georgia Benkart, On inner ideals and ad-nilpotent elements of Lie algebras, Trans. Amer. Math. Soc. 232 (1977), 61–81. MR 466242, DOI 10.1090/S0002-9947-1977-0466242-6
C. L. Chuang and P. H. Lee, Idempotents in simple rings, J. Algebra 56 (1979), no. 2, 510–515. MR 528592, DOI 10.1016/0021-8693(79)90354-5
Cristina Draper Fontanals, Antonio Fernández López, Esther García, and Miguel Gómez Lozano, The socle of a nondegenerate Lie algebra, J. Algebra 319 (2008), 2372–2394.
Antonio Fernández López, Esther García, and Miguel Gómez Lozano, The Jordan socle and finitary Lie algebras, J. Algebra 280 (2004), no. 2, 635–654. MR 2089256, DOI 10.1016/j.jalgebra.2004.06.013
Antonio Fernández López, Esther García, and Miguel Gómez Lozano, Inner ideals of finitary simple Lie algebras, J. Lie Theory 16 (2006), no. 1, 97–114. MR 2196417
Antonio Fernández López, Esther García, and Miguel Gómez Lozano, The Jordan algebras of a Lie algebra, J. Algebra 308 (2007), no. 1, 164–177. MR 2290916, DOI 10.1016/j.jalgebra.2006.02.035
—, An Artinian theory for Lie algebras, J. Algebra 319 (2008), 938–951.
Antonio Fernández López, Esther García, Miguel Gómez Lozano, and Erhard Neher, A construction of gradings of Lie algebras, Int. Math. Res. Not. IMRN 16 (2007), Art. ID rnm051, 34. MR 2353091, DOI 10.1093/imrn/rnm051
Nathan Jacobson, Lie algebras, Dover Publications, Inc., New York, 1979. Republication of the 1962 original. MR 559927
Ottmar Loos, Jordan pairs, Lecture Notes in Mathematics, Vol. 460, Springer-Verlag, Berlin-New York, 1975. MR 0444721
Ottmar Loos, On the socle of a Jordan pair, Collect. Math. 40 (1989), no. 2, 109–125 (1990). MR 1094683
Ottmar Loos and Erhard Neher, Complementation of inner ideals in Jordan pairs, J. Algebra 166 (1994), no. 2, 255–295. MR 1279257, DOI 10.1006/jabr.1994.1151
Erhard Neher, Lie algebras graded by $3$-graded root systems and Jordan pairs covered by grids, Amer. J. Math. 118 (1996), no. 2, 439–491. MR 1385288
G. B. Seligman, Modular Lie algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 40, Springer-Verlag New York, Inc., New York, 1967. MR 0245627
E. I. Zel′manov, Lie algebras with algebraic associated representation, Mat. Sb. (N.S.) 121(163) (1983), no. 4, 545–561 (Russian). MR 716113