阶化Cartan型特殊代数S（m;n）的不可约表示
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摘要
本文将Skryabin为了研究广义Witt代数的表示而提出来的(?)-模范畴理论建立在Cartan型李代数系列的特殊型李代数S(m；n)上。证明了广义限制李代数意义下的诱导模成为(?)-模范畴对象。从而决定了这类李代数所有广义p-特征高度不超过min{p~(n_i)-p~(n_(i-1))|1≤i≤m}-2的不可约模：其中在非例外权情形不可约模即为诱导模，例外权情形不可约模为诱导模的唯一商模。对于后者，通过诱导模的Koszul复形具体构造了出来，并由此确定了高度为0的所有不可约模的同构类个数，确定了所有例外权的不可约模的维数。
In this paper, we consider the (?)-module category theory in graded Cartan type special algebra S(m; n) which was firstly introduced by Skryabin to study representations of the generalized witt algebra. We prove that as a generalized restricted Lie algebra, the induced modules of S(m;n) are objects of the (?)-module category. Irreducible modules with generalized p-character no more than min are determined. In the nonexceptional case, all irreducible modules are induced modules. In the exceptional case, irrducible modules are the unique quotient modules of induced modules. For the latter case, irreducible modules are concretely constructed through the Koszul complex of induced modules. Furthermore, all isomorphism classes of irreducible modules are determined when the height of the character is 0, and the dimensions of all exceptional irreducible modules are given.
In this paper, we consider the (?)-module category theory in graded Cartan type special algebra S(m; n) which was firstly introduced by Skryabin to study representations of the generalized witt algebra. We prove that as a generalized restricted Lie algebra, the induced modules of S(m;n) are objects of the (?)-module category. Irreducible modules with generalized p-character no more than min are determined. In the nonexceptional case, all irreducible modules are induced modules. In the exceptional case, irrducible modules are the unique quotient modules of induced modules. For the latter case, irreducible modules are concretely constructed through the Koszul complex of induced modules. Furthermore, all isomorphism classes of irreducible modules are determined when the height of the character is 0, and the dimensions of all exceptional irreducible modules are given.
引文
[1] Chang Ho-Jui, Uber Wittsche Lie-Ringe, Abh. Math Sem. Univ. Hamburg 14 (1941), 151-184.
[2] Benkart G.M, Caftan subalgebras in Lie algebras of cartan type, Lie Algebras and Related Topics, CMS Conf. proc 5 (1986), 157-187.
[3] Holmes R. R., Simple modules with character height at most one for the restricted Witt algebras, J.Algebra 237 No.2(2001), 446-469.
[4] Holmes R. R. and Chaowen Zhang, Some simple modules for the restricted Cartan type Lie algebras, Journal of Pure and Applied Algebra 173(2002), 135-165.
[5] Nakano D., Projective modules over Lie algebras of Cartan type, Memoirs of AMS No.470 (1992).
[6] Shen Guang-Yu, Graded modules of graded Lie algebras of Cartan type, Ⅰ, Scientica Sinica 29(1986), 570-581.
[7] Shen Guang-Yu, Graded modules of graded Lie algebras of Cartan type, Ⅱ, Scientica Sinica 29(1986), 1009-1019.
[8] Shen Guang-Yu, Graded modules of graded Lie algebras of Cartan type, Ⅲ, Chinese Ann. Math. Ser.B 9(1988), 404-417.
[9] Skryabin, S., Independent systems of derivations and Lie algebra representations, in "Algebra and Analysis, Eds: Archrnov/Parshin/Shafarvich." Walter de Gruyter& Co, Berlin-New York, 115-150.
[10] Skryabin, S., Modular Lie algebras of Cartan type over algebraically non-closed fields, Comm. Algebra 19(1991), 1629-1741.
[11] Skryabin, S., An algebraic approach to the Lie algebras of Caftan type, Comm. Algebra 21(1993), 1229-1336.
[12] Skryabin, S., On representations of Lie algebras of derivations of commutative rings, Vestnik Moscow Univ. Ser.1 Mat.Mekh. no.4(1987), 32-35.
[13] Shu Bin, The generalized restricted representations of graded Lie algebras of cartan type, J.Algebra 194(1997), 157-177.
[14] Shu Bin, Generalized restricted Lie algebras and representations of the Zassenhaus algebra, J.Algebra 204(1998), 549-572
[15] Shu Bin, Simple generalized restricted module for graded Lie algebras of Cartan type, Chinese Sci. Bull. 43(1998), 1336-1340.
[16] Shu Bin, Conjugation in representations of the Zassenhause algebra, Acta Mathematica Sinica, Enlish series 17(2001), 319-326.
[17] 舒斌，Cartan型李代数的自同构群，数学年刊20A(1)(1999)，47-52．
[18] Shu Bin, Quasi p-mappings and representations of modular Lie algebras, in "Proceedings of the international conference on representation theorey", CHEP & Springer-Verlag, Beijing, (2000), 375-401.
[19] Shu Bin and Yao Yufeng, Irreducible representations of the generalized Jacobson-Witt algebras, Submitted.
[20] Hu Nai-Hong, The graded modules for the graded contact Cartan Lie algebras, Comm. Algebra 22(1994), 4474-4497.
[21] Hu Nai-Hong, Irreducible constituents of graded modules for graded contact Lie algebras of Cartan type, Comm. Algebra 22(1994), 5951-5970.
[22] Jiang Zhi-hong and Shen Guang-Yu, Cohomology of generalized restricted Lie algebras, J. Algebra 277(2004), 3-26.
[23] 濮燕敏，蒋志洪，特征标高度为0具有例外权的不可约H(2r，n)-模，数学年刊27A(2)(2006)，1-12．
[24] Wilson R. L., A structural characterization of the simple Lie algebras of generalized cartan type over fields of prime characteristic, J. Algebra 40(1976), 418-465.
[25] Wilson R. L., Classification of generalized witt algebras over algebraically closed filds, Trans. Amer. Math. Soc. Vol.153(1971), 191-210.
[26] Wilson R. L., Automorphisms of graded Lie algebras of Cartan type, Comm.Algebra 3 No. 7(1975), 591-613.
[27] R.Ree, On generalized witt algebras, Trans. Amer. Math. Soc. 83(1956), 510-546.
[28] Mil'ner.A., The irreducible representations of the Zassenhaus algebra, Uspekhi Mat. Nauk30 no.6(1975), 178.
[29] Strade H., Representations of the Witt algebra, J.Algebra 49(1977), 595-605.
[30] Jantzen, Representations of the Witt-Jacobson algebras in prime characteristic, talk in "The 6th international conference on representation theory of algebaic groups and quantum groups 06, Nagoya University."
[31] Blattner R.J., Induced and produced representations of Lie algebras, Trans. Amer. Math. Soc. 144(1969), 457-474.
[32] Jacobson N., Lie algebras, Interscience, New York, 1962
[33] Strade H. and Farnsteiner R., Modular Lie algebras and their representations, Marcel Dekker, New York, 1988
[2] Benkart G.M, Caftan subalgebras in Lie algebras of cartan type, Lie Algebras and Related Topics, CMS Conf. proc 5 (1986), 157-187.
[3] Holmes R. R., Simple modules with character height at most one for the restricted Witt algebras, J.Algebra 237 No.2(2001), 446-469.
[4] Holmes R. R. and Chaowen Zhang, Some simple modules for the restricted Cartan type Lie algebras, Journal of Pure and Applied Algebra 173(2002), 135-165.
[5] Nakano D., Projective modules over Lie algebras of Cartan type, Memoirs of AMS No.470 (1992).
[6] Shen Guang-Yu, Graded modules of graded Lie algebras of Cartan type, Ⅰ, Scientica Sinica 29(1986), 570-581.
[7] Shen Guang-Yu, Graded modules of graded Lie algebras of Cartan type, Ⅱ, Scientica Sinica 29(1986), 1009-1019.
[8] Shen Guang-Yu, Graded modules of graded Lie algebras of Cartan type, Ⅲ, Chinese Ann. Math. Ser.B 9(1988), 404-417.
[9] Skryabin, S., Independent systems of derivations and Lie algebra representations, in "Algebra and Analysis, Eds: Archrnov/Parshin/Shafarvich." Walter de Gruyter& Co, Berlin-New York, 115-150.
[10] Skryabin, S., Modular Lie algebras of Cartan type over algebraically non-closed fields, Comm. Algebra 19(1991), 1629-1741.
[11] Skryabin, S., An algebraic approach to the Lie algebras of Caftan type, Comm. Algebra 21(1993), 1229-1336.
[12] Skryabin, S., On representations of Lie algebras of derivations of commutative rings, Vestnik Moscow Univ. Ser.1 Mat.Mekh. no.4(1987), 32-35.
[13] Shu Bin, The generalized restricted representations of graded Lie algebras of cartan type, J.Algebra 194(1997), 157-177.
[14] Shu Bin, Generalized restricted Lie algebras and representations of the Zassenhaus algebra, J.Algebra 204(1998), 549-572
[15] Shu Bin, Simple generalized restricted module for graded Lie algebras of Cartan type, Chinese Sci. Bull. 43(1998), 1336-1340.
[16] Shu Bin, Conjugation in representations of the Zassenhause algebra, Acta Mathematica Sinica, Enlish series 17(2001), 319-326.
[17] 舒斌，Cartan型李代数的自同构群，数学年刊20A(1)(1999)，47-52．
[18] Shu Bin, Quasi p-mappings and representations of modular Lie algebras, in "Proceedings of the international conference on representation theorey", CHEP & Springer-Verlag, Beijing, (2000), 375-401.
[19] Shu Bin and Yao Yufeng, Irreducible representations of the generalized Jacobson-Witt algebras, Submitted.
[20] Hu Nai-Hong, The graded modules for the graded contact Cartan Lie algebras, Comm. Algebra 22(1994), 4474-4497.
[21] Hu Nai-Hong, Irreducible constituents of graded modules for graded contact Lie algebras of Cartan type, Comm. Algebra 22(1994), 5951-5970.
[22] Jiang Zhi-hong and Shen Guang-Yu, Cohomology of generalized restricted Lie algebras, J. Algebra 277(2004), 3-26.
[23] 濮燕敏，蒋志洪，特征标高度为0具有例外权的不可约H(2r，n)-模，数学年刊27A(2)(2006)，1-12．
[24] Wilson R. L., A structural characterization of the simple Lie algebras of generalized cartan type over fields of prime characteristic, J. Algebra 40(1976), 418-465.
[25] Wilson R. L., Classification of generalized witt algebras over algebraically closed filds, Trans. Amer. Math. Soc. Vol.153(1971), 191-210.
[26] Wilson R. L., Automorphisms of graded Lie algebras of Cartan type, Comm.Algebra 3 No. 7(1975), 591-613.
[27] R.Ree, On generalized witt algebras, Trans. Amer. Math. Soc. 83(1956), 510-546.
[28] Mil'ner.A., The irreducible representations of the Zassenhaus algebra, Uspekhi Mat. Nauk30 no.6(1975), 178.
[29] Strade H., Representations of the Witt algebra, J.Algebra 49(1977), 595-605.
[30] Jantzen, Representations of the Witt-Jacobson algebras in prime characteristic, talk in "The 6th international conference on representation theory of algebaic groups and quantum groups 06, Nagoya University."
[31] Blattner R.J., Induced and produced representations of Lie algebras, Trans. Amer. Math. Soc. 144(1969), 457-474.
[32] Jacobson N., Lie algebras, Interscience, New York, 1962
[33] Strade H. and Farnsteiner R., Modular Lie algebras and their representations, Marcel Dekker, New York, 1988