A construction of modular representations of classical Lie algebras

Authors:
Karl M. Peters and Zhiyong Shi

Journal:
Proc. Amer. Math. Soc. **122** (1994), 399-407

MSC:
Primary 17B10; Secondary 17B50

MathSciNet review:
1233981

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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.

**[B]**S. Berman,*On the construction of simple Lie algebras*, J. Algebra**27**(1973), 158–183. MR**0354793****[BFL]**D. J. Britten, V. M. Futorny, and F. W. Lemire,*Simple finite dimensional**modules*(to appear).**[BL1]**D. J. Britten and F. W. Lemire,*Irreducible representations of 𝐴_{𝑛} with a 1-dimensional weight space*, Trans. Amer. Math. Soc.**273**(1982), no. 2, 509–540. MR**667158**, 10.1090/S0002-9947-1982-0667158-4**[BL2]**D. J. Britten and F. W. Lemire,*A classification of simple Lie modules having a 1-dimensional weight space*, Trans. Amer. Math. Soc.**299**(1987), no. 2, 683–697. MR**869228**, 10.1090/S0002-9947-1987-0869228-9**[D]**Jacques Dixmier,*Enveloping algebras*, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. North-Holland Mathematical Library, Vol. 14; Translated from the French. MR**0498740****[F]**S. L. Fernando,*Lie algebra modules with finite-dimensional weight spaces. I*, Trans. Amer. Math. Soc.**322**(1990), no. 2, 757–781. MR**1013330**, 10.1090/S0002-9947-1990-1013330-8**[FP1]**Eric M. Friedlander and Brian J. Parshall,*Modular representation theory of Lie algebras*, Amer. J. Math.**110**(1988), no. 6, 1055–1093. MR**970120**, 10.2307/2374686**[FP2]**Eric M. Friedlander and Brian J. Parshall,*Deformations of Lie algebra representations*, Amer. J. Math.**112**(1990), no. 3, 375–395. MR**1055649**, 10.2307/2374747**[FP3]**E. M. Friedlander and B. J. Parshall,*Induction, deformation, and specialization of Lie algebra representations*, Math. Ann.**290**(1991), no. 3, 473–489. MR**1116233**, 10.1007/BF01459255**[KW]**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), 135-151.**[P]**Karl M. Peters,*Characters of modular torsion free representations of classical Lie algebras*, Comm. Algebra**22**(1994), no. 12, 4807–4826. MR**1285711**, 10.1080/00927879408825106**[S]**Zhiyong Shi,*Classification of pointed weak torsion free representations for classical Lie algebras*, J. Algebra**171**(1995), no. 3, 677–699. MR**1315918**, 10.1006/jabr.1995.1033**[SF]**Helmut Strade and Rolf Farnsteiner,*Modular Lie algebras and their representations*, Monographs and Textbooks in Pure and Applied Mathematics, vol. 116, Marcel Dekker, Inc., New York, 1988. MR**929682****[WK]**B. Ju. Veĭsfeĭler and V. G. Kac,*The irreducible representations of Lie 𝑝-algebras*, Funkcional. Anal. i Priložen.**5**(1971), no. 2, 28–36 (Russian). MR**0285575****[Z]**Hans Zassenhaus,*The representations of Lie algebras of prime characteristic*, Proc. Glasgow Math. Assoc.**2**(1954), 1–36. MR**0063359**

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DOI:
https://doi.org/10.1090/S0002-9939-1994-1233981-4

Article copyright:
© Copyright 1994
American Mathematical Society