A construction of modular representations of classical Lie algebras
Authors:
Karl M. Peters and Zhiyong Shi
Journal:
Proc. Amer. Math. Soc. 122 (1994), 399407
MSC:
Primary 17B10; Secondary 17B50
MathSciNet review:
1233981
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Abstract 
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Abstract: In this paper, we construct and analyze new classes of modular representations of classical Lie algebras of type C and type A. These representations include a class of pointed torsion free representations, a class of irreducible nonrestricted representations, and a class of indecomposable representations of arbitrary high dimension. The construction is based on the realization of these Lie algebras in the modular Weyl algebras acting on truncated polynomial algebras. We also classify all the irreducible representations of the modular Weyl algebra.
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 J. Dixmier, Enveloping algebras, NorthHolland, Amsterdam, 1977. MR 0498740 (58:16803b)
 [F]
 S. L. Fernando, Lie algebra modules with finite dimensional weight spaces. I, Trans. Amer. Math. Soc. 322 (1990), 757781. MR 1013330 (91c:17006)
 [FP1]
 Eric M. Friedlander and Brian J. Parshall, Modular representation theory of Lie algebras, Amer. J. Math. 110 (1988), 10551094. MR 970120 (89j:17015)
 [FP2]
 , Deformations of Lie algebra representations, Amer. J. Math. 112 (1990), 375395. MR 1055649 (91e:17012)
 [FP3]
 , Induction, deformation, and specialization of Lie algebra representations, Math. Ann. 290 (1991), 473489. MR 1116233 (92h:17021)
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 V. G. Kac and B. Weisfeiler, Cojoint action of a semisimple algebraic group and the center of the enveloping algebra in characteristic p, Indag. Math. 38 (1976), 135151.
 [P]
 Karl M. Peters, Characters of modular torsion free representations of classical Lie algebras, Comm. Algebra (to appear). MR 1285711 (95h:17010)
 [S]
 Zhiyong Shi, Classification of pointed weak torsion free representations for classical Lie algebras, J. Algebra (to appear). MR 1315918 (96b:17022)
 [SF]
 H. Strade and R. Farnsteiner, Modular Lie algebras and their representations, Marcel Dekker, New York, 1988. MR 929682 (89h:17021)
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 B. Weisfeiler and V. Kac, The irreducible representations of Lie palgebras, Functional Anal. Appl. 5 (1971), 471503. MR 0285575 (44:2793)
 [Z]
 H. Zassenhaus, The representations of Lie algebras of prime characteristic, Proc. Glasgow Math. Assoc. 2 (1954), 136. MR 0063359 (16:108c)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199412339814
PII:
S 00029939(1994)12339814
Article copyright:
© Copyright 1994
American Mathematical Society
