Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-11T03:59:23.015Z Has data issue: false hasContentIssue false
  • English
  • Français

Derivations and Invariant Forms of Lie Algebras Graded by Finite Root Systems

Published online by Cambridge University Press:  20 November 2018

Georgia Benkart
Georgia Benkart*
Affiliation:
Department of Mathematics University of Wisconsin Madison, Wisconsin 53706 U.S.A.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Lie algebras graded by finite reduced root systems have been classified up to isomorphism. In this paper we describe the derivation algebras of these Lie algebras and determine when they possess invariant bilinear forms. The results which we develop to do this are much more general and apply to Lie algebras that are completely reducible with respect to the adjoint action of a finite-dimensional subalgebra.

Keywords

17B2017B7017B25
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 50 , Issue 2 , 01 April 1998 , pp. 225 - 241
Copyright
Copyright © Canadian Mathematical Society 1998

References

[AABGP] Allison, B.N., Azam, S., Berman, S.,Gao, Y. and Pianzola, A., Extended Affine Lie Algebras and Their Root Systems. Mem. Amer.Math. Soc. 603, Providence, 1997.Google Scholar
[ABG1] Allison, B.N., Benkart, G. and Gao, Y., Central extensions of Lie algebras graded by finite root systems. submitted.Google Scholar
[ABG2] Allison, B.N., Lie algebras graded by the root system BCr, r ≥ 2. to appear.Google Scholar
[BM] Benkart, G.M. and Moody, R.V., Derivations, central extensions, and affine Lie algebras. Algebras Groups Geom. 3(1986), 456–492.Google Scholar
[BO] Benkart, G.M. and Osborn, J.M., Derivations and automorphisms of nonassociative matrix algebras. Trans. Amer. Math. Soc. 263(1981), 411–430.Google Scholar
[BS] Benkart, G. and Smirnov, O., Lie algebras graded by the root system BC1. to appear.Google Scholar
[BZ1] Benkart, G. and Zelmanov, E., Lie Algebras graded by root systems. Proc. of the Third International Conference on Nonassociative Algebras and its Applications, (ed. onzález, S.), Kluwer Publ., 1994. 31– 38.Google Scholar
[BZ2] Benkart, G. and Zelmanov, E., Lie algebras graded by finite root systems and intersection matrix algebras. Invent. Math. 126(1996), 1–45.Google Scholar
[BGK] Berman, S.,Gao, Y. and Krylyuk, Y., Quantum tori and the structure of elliptic quasi-simple Lie algebras. J. Funct. Anal 135(1996), 339–389.Google Scholar
[BGKN] Berman, S., Gao, Y., Krylyuk, Y. and Neher, E., The alternative torus and the structure of elliptic quasisimple Lie algebras of type A2. Trans. Amer.Math. Soc. 347(1995), 4315–4363.Google Scholar
[BeM] Berman, S. and Moody, R.V., Lie algebras graded by finite root systems and the intersection matrix algebras of Slodowy. Invent. Math 108(1992), 323–347.Google Scholar
[FH] Fulton, W. and Harris, J., Representation Theory.Graduate Texts In Math. 129, Springer-Verlag,NewYork, 1991.Google Scholar
[G] Gao, Y., Steinberg unitary Lie algebras and skew-dihedral homology. J. Algebra 179(1996), 261–304.Google Scholar
[Ga] Garland, H., The arithmetic theory of loop groups. Publ. Math. IHES 52(1980), 5–136.Google Scholar
[GF] Gel’fand, I.M. and Fuks, D.B., Cohomologies of the Lie algebra of vector fields on the circle. (Russian) Funktsional. Anal. i Prilozhen 2(1968), 92–93. English, Functional Anal. Appl. 2(1968), 342–343.Google Scholar
[HT] Høegh-Krohn, R. and Torressani, B.,Classification and construction of quasi-simple Lie algebras. J. Funct. Anal. 89(1990), 106–134.Google Scholar
[J] Jacobson, N., Exceptional Lie Algebras. Lecture Notes in Pure and Applied Math., Marcel Dekker, New York, 1971.Google Scholar
[K] Kac, V.G., Infinite Dimensional Lie Algebras. Third Ed., Cambridge Univ. Press, Cambridge, 1990.Google Scholar
[KKLW] Kac, V.G., Kazhdan, D., Lepowsky, J. and Wilson, R.L., Realization of the basic representation of the Euclidean Lie algebras. Adv. in Math. 42(1981), 83–112.Google Scholar
[MP] Moody, R.V. and Pianzola, A., Lie Algebras with Triangular Decompositions. Canad. Math. Soc. Series of Monographs and Adv. Texts, Wiley Interscience, New York, 1995.Google Scholar
[N] Neher, E., Lie algebras graded by 3-graded root systems. Amer. J. Math. 118(1996), 439–491.Google Scholar
[S] Seligman, G.B., Rational Methods in Lie Algebras. Lect. Notes in Pure and Applied Math. 17, Marcel Dekker, New York, 1976.Google Scholar
[Sl] Slodowy, P., Beyond Kac-Moody algebras and inside. In: Lie Algebras and Related Topics, Canad. Math. Soc. Conf. Proc. 5, (eds. Britten, Lemire, Moody), 1986. 361–371.Google Scholar
[W] Wilson, R.L., Euclidean Lie algebras are universal central extensions. In: Lie Algebras and Related Topics, (ed. David Winter), Lect. Notes in Math. 933, Springer-Verlag, New York, 1982. 210–213.Google Scholar