On Cartan subalgebras and Cartan subspaces of nonsymmetric pairs of Lie algebras
Author:
Boris Širola
Journal:
Proc. Amer. Math. Soc. 141 (2013), 22332243
MSC (2010):
Primary 17B05; Secondary 17B20, 17B22
Published electronically:
February 27, 2013
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Abstract: Let be a pair of Lie algebras, defined over a field of characteristic zero, where is semisimple and is a subalgebra reductive in . We prove a result giving a necessary and sufficient technical condition so that the following holds: ( ) For any Cartan subalgebra there exists a unique Cartan subalgebra containing . Next we study a class of pairs , satisfying ( ), which we call Cartan pairs. For such pairs and the corresponding Cartan subspaces, we prove some useful results that are classical for symmetric pairs. Thus we extend a part of the previous research on Cartan subspaces done by Dixmier, Lepowsky and McCollum.
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 A. W. Knapp, Lie Groups beyond an Introduction, Second edition, Progress in Math., vol. 140, Birkhäuser, BostonBaselBerlin, 2002. MR 1920389 (2003c:22001)
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 J. Lepowsky and G. W. McCollum, Cartan subspaces of symmetric Lie algebras, Trans. Amer. Math. Soc. (1976), 217228. MR 0404361 (53:8163)
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 T. Levasseur and S. P. Smith, Primitive ideals and nilpotent orbits in type , J. Algebra (1988), 81105. MR 931902 (89f:17013)
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 , Pairs of semisimple Lie algebras and their maximal reductive subalgebras, Algebr. Represent. Theory (2008), 233250. MR 2403292 (2009d:17017)
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 , Normalizers and selfnormalizing subgroups, Glas. Mat. Ser. III 46 (2011), 385414. MR 2855021
 [Š5]
 , Normalizers, distinguished nilpotent orbits and compatible pairs of Borel subalgebras, in preparation.
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 , Noncommutative algebras and unitary representations, Proc. Sympos. Pure Math., vol. 48, Amer. Math. Soc., Providence, RI, 1988, pp. 3560. MR 974331 (90a:22015)
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Additional Information
Boris Širola
Affiliation:
Department of Mathematics, University of Zagreb, Bijenička 30, 10000 Zagreb, Croatia
Email:
sirola@math.hr
DOI:
http://dx.doi.org/10.1090/S00029939201311508X
PII:
S 00029939(2013)11508X
Keywords:
Pair of Lie algebras,
semisimple Lie algebra,
reductive subalgebra,
Cartan subalgebra,
Cartan subspace,
Cartan pair.
Received by editor(s):
December 2, 2010
Received by editor(s) in revised form:
September 16, 2011, and October 11, 2011
Published electronically:
February 27, 2013
Additional Notes:
The author was supported in part by the Ministry of Science, Education and Sports, Republic of Croatia, Grant No. 900194134.
Communicated by:
Gail R. Letzter
Article copyright:
© Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
