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PARABOLIC SUBROOT SYSTEMS AND THEIR APPLICATIONS

Part of: Global differential geometry Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  30 April 2019

JOHN M. BURNS  and
MOHAMMAD A. MAKROONI
JOHN M. BURNS
Affiliation:
School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland e-mails: [email protected], [email protected]
MOHAMMAD A. MAKROONI
Affiliation:
School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland e-mails: [email protected], [email protected]
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Abstract

In this note we consider parabolic subroot systems of a complex simple Lie Algebra. We describe root theoretic data of the subroot systems in terms of that of the root system and we give a selection of applications of our results to the study of generalized flag manifolds.

MSC classification

Primary: 53C30: Homogeneous manifolds
Secondary: 17B20: Simple, semisimple, reductive (super)algebras
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 62 , Issue 2 , May 2020 , pp. 355 - 366
Copyright
Copyright © Glasgow Mathematical Journal Trust 2019

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References

Akhiezer, D. N., Lie group actions in complex analysis, Aspects of Mathematics, E27, (Friedr. Vieweg & Sohn, Braunschweig, 1995).CrossRefGoogle Scholar
Alekseevsky, D. V. and Perelomov, A. M., Invariant Kähler-Einstein metrics on compact homogeneous spaces, Funct. Anal. Appl. 20 (1986), 171–182.Google Scholar
Arvanitoyeorgos, A. and Chrysikos, I., Invariant Einstein metrics on flag manifolds with four isotropy summands, Ann. Global Anal. Geom. 37(2) (2010), 185–219.10.1007/s10455-009-9183-7CrossRefGoogle Scholar
Beltrametti, M. C., Fania, M. L. and Sommese, A. J., On the discriminant variety of a projective manifold, Forum Math. 4(6) (1992), 529–547.Google Scholar
Borel, A., On the curvature tensor of the Hermitian symmetric manifolds, Ann. Math. 71(2) (1960), 508–521.10.2307/1969940CrossRefGoogle Scholar
Borel, A. and de Siebenthal, J., Les sous-groupes fermés de rang maximum des groupes de Lie clos, Comment. Math. Helv. 23 (1949), 200–221.Google Scholar
Borel, A. and Hirzebruch, F., Characteristic classes and homogeneous spaces. I, Am. J. Math. 80 (1958), 458–538.Google Scholar
Bourbaki, N., Group et algèbres de Lie. Ch. 4, 5 et 6, (Hermann, Paris, 1968).Google Scholar
Burns, J. M. and Clancy, M. J., Weight sum formulae in Lie algebra representations, J. Algebra 257(1) (2002), 1–12.Google Scholar
Burns, J. M. and Clancy, M. J., Recurrence relations, Dynkin diagrams and Plcker formulae, Glasg. Math. J. 49(1) (2007), 53–59.Google Scholar
Burns, J. M. and Makrooni, M. A., Compact homogeneous spaces with positive Euler characteristic and their ‘Strange Formula’, Quart. J. Math. 66 (2015), 507–516.10.1093/qmath/hav009CrossRefGoogle Scholar
Carles, R., Méthode récurrente pour la classification des systèmes de racines réduits et irréductibles, C. R. Acad. Sci. Paris Sér, A–B 276 (1973), A355–A358.Google Scholar
Carles, R., Dimensions des représentations fondamentales des algébres de Lie de type G 2, F 4, E 6, E 7, E 8, C. R. Acad. Sci. Paris Sér. A–B 276 (1973), A451–A453.Google Scholar
Cellini, P. and Marietti, M., Root polytopes and Borel subalgebras, Int. Res. Not. 12(12) (2015), 4392–4420.Google Scholar
Chen, B. Y. and Nagano, T., Totally geodesic submanifolds of symmetric spaces. II, Duke Math. J. 45(2) (1978), 405–425.Google Scholar
Fino, A. and Salamon, S. M., Observations on the topology of symmetric spaces. Geometry and physics (Aarhus, 1995), Lecture Notes in Pure and Appl. Math., vol. 184, (Dekker, New York, 1997), 275–286.Google Scholar
Helgason, S., Differential geometry, Lie groups, and symmetric spaces, (Academic Press, New York, 1978).Google Scholar
Knop, F. and Menzel, G., Duale Varietten von Fahnenvarietten, Comment. Math. Helv. 62(1) (1987), 38–61.10.1007/BF02564437CrossRefGoogle Scholar
Macdonald, I. G., Affine root systems and Dedekind’s η-function, Invent. Math. 15 (1972), 91–143CrossRefGoogle Scholar
Quast, P., Centrioles in symmetric spaces, Nagoya Math. J. 211 (2013), 51–77.CrossRefGoogle Scholar
Snow, D. M., The nef value and defect of homogeneous line bundles, Trans. Am. Math. Soc. 340(1) (1993), 227–241.10.1090/S0002-9947-1993-1144015-8CrossRefGoogle Scholar
Snow, D. M., Nef value of homogeneous line bundles and related vanishing theorems, Forum Math. 7(3) (1995), 385–392.10.1515/form.1995.7.385CrossRefGoogle Scholar
Suter, R., Coxeter and dual Coxeter numbers, Comm. Algebra 26(1) (1998), 147–153.CrossRefGoogle Scholar
Tamaru, H., On certain subalgebras of graded Lie algebras, YokohamaMath. J. 46(2) (1999), 127–138.Google Scholar
Wolf, J. A., Spaces of Constant Curvature, 5th edn., (Publish or Perish Inc., Wilmington, DE, 1984).Google Scholar