On Cartan subalgebras and Cartan subspaces of nonsymmetric pairs of Lie algebras

Author:
Boris Širola

Journal:
Proc. Amer. Math. Soc. **141** (2013), 2233-2243

MSC (2010):
Primary 17B05; Secondary 17B20, 17B22

DOI:
https://doi.org/10.1090/S0002-9939-2013-11508-X

Published electronically:
February 27, 2013

MathSciNet review:
3043005

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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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Additional Information

**Boris Širola**

Affiliation:
Department of Mathematics, University of Zagreb, Bijenička 30, 10000 Zagreb, Croatia

Email:
sirola@math.hr

DOI:
https://doi.org/10.1090/S0002-9939-2013-11508-X

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. 900-194134.

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.