On Cartan subalgebras and Cartan subspaces of nonsymmetric pairs of Lie algebras
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- by Boris Širola
- Proc. Amer. Math. Soc. 141 (2013), 2233-2243
- DOI: https://doi.org/10.1090/S0002-9939-2013-11508-X
- Published electronically: February 27, 2013
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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.References
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Bibliographic Information
- Boris Širola
- Affiliation: Department of Mathematics, University of Zagreb, Bijenička 30, 10000 Zagreb, Croatia
- Email: sirola@math.hr
- 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
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 3043005