Completable nilpotent Lie superalgebras
Mingzhong WU
Completable nilpotent Lie superalgebras
We discuss a class of filiform Lie superalgebras Ln,m. From these Lie superalgebras, all the other filiform Lie superalgebras can be obtained by deformations. We have decompositions of (Ln,m) and (Ln,m). By computing a maximal torus on each Ln,m, we show that Ln,m are completable nilpotent Lie superalgebras. We also view Ln,m as Lie algebras, prove that Ln,m are of maximal rank, and show that Ln,m are completable nilpotent Lie algebras. As an application of the results, we show a Heisenberg superalgebra is a completable nilpotent Lie superalgebra.
Filiform Lie superalgebra / Heisenberg superalgebra / completable nilpotent Lie superalgebra / maximal torus / complete Lie superalgebra
[1] |
Bermúdez J M A, Campoamor R. Completable filiform Lie algebras. Linear Algebra Appl, 2003, 367: 185-191
CrossRef
Google scholar
|
[2] |
Bordemann M, Gómez J R, Khakimdjanov Y, Navarro R M. Some deformations of nilpotent Lie superalgebras. J Geom Phys, 2007, 57: 1391-1403
CrossRef
Google scholar
|
[3] |
Bourbaki N. Groupes et Algèbres de Lie. Chap. I. Paris: Hermann, 1960
|
[4] |
Chun J H, Lee J S. On complete Lie superalgebras. Commun Korean Math Soc, 1996, 2: 323-334
|
[5] |
Gómez J R, Khakimdjanov Y, Navarro R M. Some problems concerning to nilpotent Lie superalgebras. J Geom Phys, 2004, 51: 473-486
CrossRef
Google scholar
|
[6] |
Gómez J R, Khakimdjanov Y, Navarro R M. Infinitesimal deformations of the Lie superalgebra Ln,m. J Geom Phys, 2008, 58: 849-859
CrossRef
Google scholar
|
[7] |
Jacobson N. Lie Algebras. New York: Wiley Interscience, 1962
|
[8] |
Jiang C P, Meng D J, Zhang S Q. Some complete Lie algebras. J Algebra, 1996, 186: 807-817
CrossRef
Google scholar
|
[9] |
Kac V G. Lie superalgebras. Adv Math, 1977, 26: 8-96
CrossRef
Google scholar
|
[10] |
Khakimdjanov Y, Navarro R M. A complete description of all the infinitesimal deformations of the Lie superalgebra Ln,m. J Geom Phys, 2010, 60: 131-141
CrossRef
Google scholar
|
[11] |
Meng D J. Some results on complete Lie algebras. Comm Algebra, 1994, 22: 5457-5507
CrossRef
Google scholar
|
[12] |
Meng D J, Zhu L S. Solvable complete Lie algebras I. Comm Algebra, 1996, 24: 4187-4197
|
[13] |
Mostow G D. Fully reducible subgroups of algebaic groups. Amer J Math, 1956, 78: 200-221
CrossRef
Google scholar
|
[14] |
Ren B, Meng D J. Some 2-step nilpotent Lie algebras I. Linear Algebra Appl, 2001, 338: 77-98
CrossRef
Google scholar
|
[15] |
Santharoubane L J. Kac-Moody Lie algebras and the classification of nilpotent Lie algebras of maximal rank. Canad J Math, 1982, 34: 1215-1239
CrossRef
Google scholar
|
[16] |
Schenkman E V. A theory of subinvariant Lie algebras. Amer J Math, 1951, 73: 453-474
CrossRef
Google scholar
|
[17] |
Vergne M. Cohomologie des algèbres de Lie nilpotentes, Application à l’étude de la variété des algèbres de Lie nilpotentes. Bull Soc Math France, 1970, 98: 81-116
|
[18] |
Wang L Y, Meng D J. Some results on complete Lie superalgebras. Linear Algebra Appl, 2002, 355: 1-14
CrossRef
Google scholar
|
[19] |
Wang L Y, Meng D J. Some complete Lie superalgebras. Linear Algebra Appl, 2003, 369: 339-349
CrossRef
Google scholar
|
[20] |
Zhu L S, Meng D J. The classification of complete Lie algebras with low dimensions. Algebra Colloq, 1997, 4: 95-109
|
/
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