Support varieties of semisimple-character representations for Cartan type Lie algebras

Abstract

In this paper, we study the support varieties for Lie algebras L of Cartan type. We give some description for the support varieties of any finitedimensional L-module with character χ, whenever the height of the character χ is not too large. And a more concrete computation can be made for a class of modules with semisimple characters.

Keywords

Lie algebra of Cartan type / generalized restricted Lie algebra / support variety / semisimple character / exceptional module

References

[1] Feldvoss J., Nakano D. Representation theory of the Witt algebra. J Algebra, 1998, 203, 447-469

[2] Friedlander E., Parshall B. Geometry of p-unipotent Lie algebras. J Algebra, 1987, 109, 25-45

[3] Friedlander E., Parshall B. Modular representation theory of Lie algebras. Am J Math, 1988, 110, 1055-1094

[4] Hochschild G. Cohomology of restricted Lie algebras. Am J Math, 1954, 76, 555-580

[5] Holmes R. R. Simple modules with character height at most one for the restricted Witt algebras. J Algebra, 2001, 237 2 446-469

[6] Holmes R. R., Zhang C. W. Some simple modules for the restricted Cartan-type Lie algebras. J Pure Appl Algebra, 2002, 173, 135-165

[7] Hu N. H. The graded modules for the graded contact Cartan algebras. Commun Algebra, 1994, 22 11 4475-4497

[8] Jantzen J. C. Kohomologie von p-Lie algebren und nilpotente Elemente. Abh Math Sem Univ Hamburg, 1986, 56, 191-219

[9] Kostrikin A. I., Šafarevič I. R. Graded Lie algebras of finite characteristic. Math USSR Izv, 1969, 3, 237-304

[10] Lin Z. Z., Nakano D. Algebraic group actions in the cohomology theory of Lie algebras of Cartan type. J Algebra, 1996, 179, 852-888

[11] Premet A. Irreducible representations of Lie algebras of reductive groups and the Kac-Weisfeiler conjecture. Invent Math, 1995, 121, 79-117

[12] Premet A. Support varieties of non-restricted modules over Lie algebras of reductive groups. J London Math Soc, 1997, 55 2 236-250

[13] Shen G. Y. Graded modules of graded Lie algebras of Cartan type III. Chinese Ann Math, Ser B, 1988, 9, 404-417

[14] Shu B. The realizations of primitive p-envelopes and the support varieties for graded Cartan type Lie algebras. Commun Algebra, 1997, 25 10 3209-3223

[15] Shu B, Yao Y F. Irreducible modules of the generalized Jacobson-Witt algebras. Algebra Colloquium (to appear)

[16] Strade H., Farnsteiner R. Modular Lie Algebras and Their Representations, 1988, New York: Marcel Dekker.

[17] Wilson R. L. Automorphisms of graded Lie algebras of Cartan type. Commun Algebra, 1975, 3 7 591-613

[18] Yao Y. F., Shu B. Irreducible representations of the special algebras in prime characteristic. Contemp Math, 2009, 478, 273-295

[19] Yao Y F, Shu B. Irreducible representations of the Hamiltonian algebra H(2r;n). J Austral Math Soc (to appear)

[20] Zhang C. W. Representations of the restricted Lie algebras of Cartan-type. J Algebra, 2005, 290, 408-432