Tilting modules for the nonrestricted representations of modular Lie algebra
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摘要: 令
$ G $ 为素特征$ p $ 的代数闭域$ k $ 上连通的简约代数群. 李代数$ {\frak {g}} = {\rm{Lie}}(G) $ ,$ U_{\chi}({\frak {g}}) $ 是$ {\frak {g}} $ 的约化包络代数. 在$ p $ -特征$ \chi $ 具有标准 Levi 型时, 证明了一个$ U_{\chi}({\frak {g}}) $ -模$ Q $ 是倾斜模的充分必要条件是$ Q $ 是投射模.Abstract: Let$ G $ be a connected reductive algebraic group over an algebraically closed field$ k $ of prime characteristic$ p $ , and let$ {\frak {g}} = {\rm{Lie}}(G) $ ,$U_{\chi}({\frak {g}}) $ be the reduced enveloping algebra. In this paper, when$ p $ -character$ \chi $ has the standard Levi form, we prove that a$ U_{\chi}({\frak {g}}) $ -module$ Q $ is a tilting module if and only if it is projective.-
Key words:
- tilting module /
- standard Levi form /
- projective module /
- nonrestricted representation
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[1] JANTZEN J C. Representations of Lie algebras in prime characteristic [M]//Representation Theories and Algebraic Geometry. Dordrecht: Kluwer, Academic Publishers, 1998: 185-235. [2] KAC V, WEISFEILER B. Coadjoint action of a semisimple algebraic group and the center of the enveloping algebra in characteristic p [J]. Indag Math, 1976, 38: 136-151. [3] FRIEDLANDER E M, PARSHALL B. Modular representation theory of Lie algebras [J]. Amer J Math, 1988, 110: 1055-1093. doi: 10.2307/2374686 [4] FRISK A. Dlab’s theorem and tilting modules for stratified algebras [J]. Journal of Algebra, 2007, 314: 507. doi: 10.1016/j.jalgebra.2006.08.041 [5] CLINE E, PARSHALL B, SCOTT L. On injective modules for infinitesimal algebraic groups I [J]. Journal of the London Mathematical Society, 1985, 31(2): 277-291. [6] LI Y Y, SHU B. Filtrations in modular representations of reductive Lie algebras [J]. Algebra Colloq, 2010, 17: 265-282. doi: 10.1142/S1005386710000283 [7] ÁGOSTON I, HAPPLE D, LUKÁCS E, et al. Standardly stratified algebras and tilting [J]. Journal of Algebra, 2000, 226: 144-160. doi: 10.1006/jabr.1999.8154 [8] ANDERSEN H H, JANTZEN J C, SOERGEL W. Representations of quantum groups at a p-th root of unity and of semisimple groups in charactertic p: Independence of p [J]. Asterisque, 1994, 220: 1-320. [9] JANTZEN J C. Modular representations of reductive Lie algebras [J]. Journal of Pure and Applied Algebra, 2000, 152: 133-185. doi: 10.1016/S0022-4049(99)00142-5 [10] LI Y Y. Cohomology of reductive modular Lie algebras [J]. 华东师范大学学报 (自然科学版), 2011(5): 113-120. [11] FRIEDLANDER E M, PARSHALL B. Geometry of p-unipotent Lie algebras [J]. J Algebra, 1987, 109: 25-45. doi: 10.1016/0021-8693(87)90161-X [12] JANTZEN J C. Kohomologie von p-Lie-Algebren und nilpotente Elemente [J]. Abh Math Sem Univ Hamburg, 1986, 56: 191-219. doi: 10.1007/BF02941516