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ABELIAN IDEALS IN A COMPLEX SIMPLE LIE ALGEBRA

Part of: Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  10 June 2016

PATRICK J. BROWNE
PATRICK J. BROWNE*
Affiliation:
School of Mathematics, Statistics and Applied Mathematics, N.U.I Galway, Ireland e-mail: [email protected]
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Abstract

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In this note, we give a new simple construction of all maximal abelian ideals in a Borel subalgebra of a complex simple Lie algebra. We also derive formulas for dimensions of certain maximal abelian ideals in terms of the theory of Borel de Siebenthal.

MSC classification

Primary: 17B22: Root systems
Secondary: 17B99: None of the above, but in this section
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 59 , Issue 1 , January 2017 , pp. 255 - 264
Copyright
Copyright © Glasgow Mathematical Journal Trust 2016 

References

REFERENCES

1. Borel, A. and de Siebenthal, J., Les sous-groupes fermes de rang maximum de Lie clos, Comment. Math. Helv 23 (1949), 200–221.Google Scholar
2. Bourbaki, N., Group et Algebres de Lie. Chapitres 4, 5 et 6 (Masson, Paris, 1981).Google Scholar
3. Burns, J. M. and Clancy, M. J., Weight sum formula in Lie algebra representations, J. Algebra 257 (1) (2002), 1–12.CrossRefGoogle Scholar
4. Cellini, P. and Papi, P., Abelian ideals of Borel subalgebras and affine Weyl groups, Adv. Math. 187 (2) (2004), 320–361.CrossRefGoogle Scholar
5. Chari, V., Dolbin, R. and Ridenour, T., Ideals in parabolic subalgebras of simple Lie algebras, Contemp. Math., Amer. Math. Soc., Providence, RI, 490 (2009), 47–60.CrossRefGoogle Scholar
6. Humphreys, J. Introduction to Lie algebras and representation theory (Springer, New York, 1997).Google Scholar
7. Kostant, B., Eigenvalues of a Laplacian and commutative Lie subalgebras, Topology, 3 (2) (1965), 147–159.CrossRefGoogle Scholar
8. Kostant, B., The set of abelian ideals of a Borel subalgebra, Cartan decompositions, and discrete series representations, Internat. Math. Res. Notices 1 (5) (1998), 225–252.Google Scholar
9. Panyushev, D. and Röhrle, G. C., Spherical orbits and abelian ideals, Adv. Math. 159 (2) (2001), 229–246.Google Scholar
10. Röhrle, G., On normal abelian subgroups in parabolic groups, Ann. de l'institut Fourier 5 (5) (1998), 1455–1482.Google Scholar
11. Suter, R., Abelian ideals in a Borel subalgebra of a complex simple Lie algebra, Inventiones Math. 156 (1) (2004), 175–221.CrossRefGoogle Scholar
12. Tamaru, H., Homogeneous Einstein manifolds attached to graded Lie algebras, Mathematische Zeitschrift 259 (1) (2008), 171–186.Google Scholar
13. Wolf, J. A., Spaces of constant curvature - fifth edition., (Publish or perish, Wilmington, DE, 1984).Google Scholar