阶化Cartan型李代数W（m；n）的I（X）-诱导表示
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摘要
这篇论文主要讨论特征p(p>2)域上W(m;n)型李代数的表示。当特征x正则半单时,我们可以将其高秩的表示约化至低秩的表示。由于阶化Carton型李代数L=W(m;n)当n≠1时不是限制李代数,Kac-Weisfeiler的特征函数划分不可约模的基本方法已不适用.通过舒斌关于广义限制李代数的讨论,我们知道广义限制李代数的x-约化模范畴与其本原p-包络的(?)-约化模范畴足一致的,其中(?)为x在本原p-包络上的平凡扩张。因此我们可以通过讨论L的本原p-包络的不可约表示来给出L的不可约表示。对于一个指标集I(?){1,2,…,m}及它的补集(?){1,2,…,m}\I,我们有除幂代数(?)以及两个广义Witt代数W(I)和W(?)。对于任意的李代数(?),我们可以定义一个关于除幂代数(?)的loop代数:(?)我们可以约化(?)的表示至其阶化子代数(?)_[0],I=S_(?)W(?)(I)的表示,其中由于(?)W(I)的任意子代数被W(?)及S_(?)正规化,我们可以考虑W(I)和W(?)的诱导表示。于是当特征x正则半单时,我们将L的表示约化至低秩的表示:当x是正则半单时,W(m;n)的不可约广义x-约化表示等同于L_I=S(?)W(?)(I)相应的不可约表示的诱导表示。
In this paper, we discuss the representation of Cartan type Lie algebra W(m; n) in characteristic p > 2, from the viewpoint of reducing rank. When the character x is regular semisimple, we can reduce higer-rank representations to lower-rank representations. Because the graded Car-tan tpye Lie algebra L = W(m; n) is not a restricted Lie algebra when n ≠ 1, Kac-Weisfeiler's method of dividing the irreducible modules with character functions is not work. Thanks to the discuss of generalized restricted Lie algebras by Shu Bin, we have known that the generalized restricted x-reduced module category coincides with the x-reduced module category of its primitive p-envelope, where x|- is the trivial extension of x in the primitive p-envelope. So we can describe the irreducible representation of L = W(m; n) by discussing the irreducible representation of its primitive p-envelope For an index subset I {1,2, ... ,m} and its supplement I|^ {1,2, ... ,m} \ I, we have the divided powrer algebra (I|^) and the two generalized Witt algebras W(I) and W(I|^). For any Lie algebra g, we can define a loop algebra associated with a divided algebra g. We can reduce the representation of to the representation of its graded Lie subalgebra and S_(I|^) normalize each loop subalgebra of (I|^) W{I). This makes it possible to consider some induced representations related to W(I) and W(I|^). From this viewpoint we can reduce the representation of L to the lower-rank representation when X is semisimple: when x is semisimple,
the irreducible generalized x - reduced representation of W(m;n) coincides with the induced representation for the corresponding irreducible representation of
In this paper, we discuss the representation of Cartan type Lie algebra W(m; n) in characteristic p > 2, from the viewpoint of reducing rank. When the character x is regular semisimple, we can reduce higer-rank representations to lower-rank representations. Because the graded Car-tan tpye Lie algebra L = W(m; n) is not a restricted Lie algebra when n ≠ 1, Kac-Weisfeiler's method of dividing the irreducible modules with character functions is not work. Thanks to the discuss of generalized restricted Lie algebras by Shu Bin, we have known that the generalized restricted x-reduced module category coincides with the x-reduced module category of its primitive p-envelope, where x|- is the trivial extension of x in the primitive p-envelope. So we can describe the irreducible representation of L = W(m; n) by discussing the irreducible representation of its primitive p-envelope For an index subset I {1,2, ... ,m} and its supplement I|^ {1,2, ... ,m} \ I, we have the divided powrer algebra (I|^) and the two generalized Witt algebras W(I) and W(I|^). For any Lie algebra g, we can define a loop algebra associated with a divided algebra g. We can reduce the representation of to the representation of its graded Lie subalgebra and S_(I|^) normalize each loop subalgebra of (I|^) W{I). This makes it possible to consider some induced representations related to W(I) and W(I|^). From this viewpoint we can reduce the representation of L to the lower-rank representation when X is semisimple: when x is semisimple,
the irreducible generalized x - reduced representation of W(m;n) coincides with the induced representation for the corresponding irreducible representation of
引文
[1] B. Shu, Representations of Cartan type Lie algebras in charaeteristic p [J], J. Algebra 256 (2002), 7-27.
[2] E.M. Friedlander, B.J. Parshall, Modular representation theory of Lie algebras [J],Amer. J. Math. 110 (1988), 1055-1093.
[3] B. Shu, Generalized restricted Lie algebras and the representations of Zassenhaus algebra [J], J. Algebra 204 (1998), 549-572.
[4] B. Shu, The generalized restricted representations of graded Lie algebras of Cartan type [J], J. Algebra 194 (1997), 157-177.
[5] B. Shu, Quasi p-mappings and representations of modular Lie algebras, in: Proc.Internet. Conf. on Representation Theory, July 1998, Shanghai, China, CHEP and Springer, 2000, pp. 375-401.
[6] J.C. Jantzen,Representations of Lie algebras in prime characteristic, in: A. Broer (Ed.), Representation Theories and Algebraic Geometry, Proceedings, Montreal, 1997, in: NATO ASI Series, Vol. C514, Kluwer, Dordrecht, 1998, pp. 185-235.
[7] D.K. Nakano, Projective modules over Lie algebras of Cartan type [J], Mem. Amer.Math. Soc. 470 (1992).
[8] H. Strade, R. Farnsteiner, Modular Lie Algebras and Their Representations [M],(Monographs and Textbooks in Pure and Applied Mathematics 116), Dekker, New York, 1988.
[9] H.-J. Chang, Uber Wittsche Lie-Ringe [J], Abh. Math. Sem. Univ. Hamburg 14 (1941), 151-184.
[10] J. Feldvoss, D.K. Nakano, Representation theory of the Witt algebra [J], J. Algebra 203 (1998), 447-469,
[11] G.-Y. Shen, Graded modules of graded Lie algebras of Cartan type (J) [J], Sei. sin.29 (1986), 570-581.
[12] G.-Y.Shen, Graded modules of graded Lie algebras of Cartan type (Ⅱ) [J], Sci. sin.29 (1986), 1009-1019.
[13] G.-Y. Shen, Graded modules of graded Lie algebras of Cartan type (Ⅲ) [J], Chinese Ann. Math. D 9 (1988), 404-417.
[14] B. Shu, Reduction theorems in the cohomology of generalized restricted Lie algebras and their applications [J], Chinese Ann.Math. Ser. B 19 (1998), No.4, 421-432.
[15] B. Shu, Simple generalized restricted modules for graded Lie algebras of Cartan type [J], Chinese Sci. Bull. 43 (1998), 1336-1340.
[16] B. Shu, The realization of primitive p-envelopes and the support varieties for graded Cartan type Lie algebras [J], Comm. Algebra 25, No.10 (1997), 3209-3223.
[17] N. Jacobson, Lie Algebras [M], Interscience, New York, 1962.
[18] N. Jacobson, Basic Algebra(Ⅰ) [M], W. H. Freeman, New York, 1974.
[19] N. Jacobson, Basic Algebra(Ⅱ) [M], W. H. Freeman, New York, 1985.
[20] G.B. Seligman, Modular Lie Algebras [M], Springer-Verlag Berlin Heidelberg, New York, 1967.
[21] J.C. Jantzen, Modular representations of reductive Lie algebras [J], Journal of Pure and Applied Algebra 152 (2000), 133-185.
[22] J.E. Humphreys, Modular representations of classical Lie algebras and semisimple groups [J], J. Algebra,19 (1971), 51-79.
[23] J.E. Humphreys, Linear Algebraic Groups [M], (Graduate Texts in Mathematics 21), Springer, New York, 1975.
[24] J.E. Humphreys, Introduetion to Lie algebras and Representation Theory [M],(Graduate Texts in Mathematies 9), Springer, New York, 1972.
[25] E. Abe, Hopf Algebras [M], Cambridge Univ. Press, Cambridge, United Kingdom.1977.
[26] M. Kapranov, Yu. Manin, Modules and Morita theorem for operads [J], Amer. J. Math. 123 (2001), no.5, 811-838.
[27] E. M. Friedlander, B. J. Parshall, Deformations of Lie algebra representations [J], Amer. J. Math. 112 (1990), 375-395.
[28] V. Kac, B. Weisfeiler, Coadjoint action of a. semi-simple algebraic group and the center of the enveloping algebra in characteristic p [J], Indag. Math. 38 (1976), 136-151.
[29] V. Kac, Infinite Dimensional Lie Algebras [M], Cambridge Univ. Press, Cambridge, 1990.
[30] 孟道骥,复半单李代数引论,北京大学出版社,北京,1998.
[31] 陈志杰,代数基础,华东师范大学出版社,上海,2001.
[2] E.M. Friedlander, B.J. Parshall, Modular representation theory of Lie algebras [J],Amer. J. Math. 110 (1988), 1055-1093.
[3] B. Shu, Generalized restricted Lie algebras and the representations of Zassenhaus algebra [J], J. Algebra 204 (1998), 549-572.
[4] B. Shu, The generalized restricted representations of graded Lie algebras of Cartan type [J], J. Algebra 194 (1997), 157-177.
[5] B. Shu, Quasi p-mappings and representations of modular Lie algebras, in: Proc.Internet. Conf. on Representation Theory, July 1998, Shanghai, China, CHEP and Springer, 2000, pp. 375-401.
[6] J.C. Jantzen,Representations of Lie algebras in prime characteristic, in: A. Broer (Ed.), Representation Theories and Algebraic Geometry, Proceedings, Montreal, 1997, in: NATO ASI Series, Vol. C514, Kluwer, Dordrecht, 1998, pp. 185-235.
[7] D.K. Nakano, Projective modules over Lie algebras of Cartan type [J], Mem. Amer.Math. Soc. 470 (1992).
[8] H. Strade, R. Farnsteiner, Modular Lie Algebras and Their Representations [M],(Monographs and Textbooks in Pure and Applied Mathematics 116), Dekker, New York, 1988.
[9] H.-J. Chang, Uber Wittsche Lie-Ringe [J], Abh. Math. Sem. Univ. Hamburg 14 (1941), 151-184.
[10] J. Feldvoss, D.K. Nakano, Representation theory of the Witt algebra [J], J. Algebra 203 (1998), 447-469,
[11] G.-Y. Shen, Graded modules of graded Lie algebras of Cartan type (J) [J], Sei. sin.29 (1986), 570-581.
[12] G.-Y.Shen, Graded modules of graded Lie algebras of Cartan type (Ⅱ) [J], Sci. sin.29 (1986), 1009-1019.
[13] G.-Y. Shen, Graded modules of graded Lie algebras of Cartan type (Ⅲ) [J], Chinese Ann. Math. D 9 (1988), 404-417.
[14] B. Shu, Reduction theorems in the cohomology of generalized restricted Lie algebras and their applications [J], Chinese Ann.Math. Ser. B 19 (1998), No.4, 421-432.
[15] B. Shu, Simple generalized restricted modules for graded Lie algebras of Cartan type [J], Chinese Sci. Bull. 43 (1998), 1336-1340.
[16] B. Shu, The realization of primitive p-envelopes and the support varieties for graded Cartan type Lie algebras [J], Comm. Algebra 25, No.10 (1997), 3209-3223.
[17] N. Jacobson, Lie Algebras [M], Interscience, New York, 1962.
[18] N. Jacobson, Basic Algebra(Ⅰ) [M], W. H. Freeman, New York, 1974.
[19] N. Jacobson, Basic Algebra(Ⅱ) [M], W. H. Freeman, New York, 1985.
[20] G.B. Seligman, Modular Lie Algebras [M], Springer-Verlag Berlin Heidelberg, New York, 1967.
[21] J.C. Jantzen, Modular representations of reductive Lie algebras [J], Journal of Pure and Applied Algebra 152 (2000), 133-185.
[22] J.E. Humphreys, Modular representations of classical Lie algebras and semisimple groups [J], J. Algebra,19 (1971), 51-79.
[23] J.E. Humphreys, Linear Algebraic Groups [M], (Graduate Texts in Mathematics 21), Springer, New York, 1975.
[24] J.E. Humphreys, Introduetion to Lie algebras and Representation Theory [M],(Graduate Texts in Mathematies 9), Springer, New York, 1972.
[25] E. Abe, Hopf Algebras [M], Cambridge Univ. Press, Cambridge, United Kingdom.1977.
[26] M. Kapranov, Yu. Manin, Modules and Morita theorem for operads [J], Amer. J. Math. 123 (2001), no.5, 811-838.
[27] E. M. Friedlander, B. J. Parshall, Deformations of Lie algebra representations [J], Amer. J. Math. 112 (1990), 375-395.
[28] V. Kac, B. Weisfeiler, Coadjoint action of a. semi-simple algebraic group and the center of the enveloping algebra in characteristic p [J], Indag. Math. 38 (1976), 136-151.
[29] V. Kac, Infinite Dimensional Lie Algebras [M], Cambridge Univ. Press, Cambridge, 1990.
[30] 孟道骥,复半单李代数引论,北京大学出版社,北京,1998.
[31] 陈志杰,代数基础,华东师范大学出版社,上海,2001.