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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
Published electronically: February 27, 2013
MathSciNet review: 3043005
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Abstract: Let $ (\mathfrak{g},\mathfrak{g}_1)$ be a pair of Lie algebras, defined over a field of characteristic zero, where $ \mathfrak{g}$ is semisimple and $ \mathfrak{g}_1$ is a subalgebra reductive in $ \mathfrak{g}$. We prove a result giving a necessary and sufficient technical condition so that the following holds: ( $ \boldsymbol {\mathsf {Q}1}$) For any Cartan subalgebra $ \mathfrak{h}_1\subseteq \mathfrak{g}_1$ there exists a unique Cartan subalgebra $ \mathfrak{h}\subseteq \mathfrak{g}$ containing $ \mathfrak{h}_1$. Next we study a class of pairs $ (\mathfrak{g},\mathfrak{g}_1)$, satisfying ( $ \boldsymbol {\mathsf {Q}1}$), 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

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.