包络超代数中心的Azumaya特征以及幂零锥的不可约性
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摘要
本文研究了特征为素数的代数闭域上的基本典型李超代数和Cartan型李代数的一些结构和表示理论.本文的主要研究成果有下面几个方面：
第一部分刻画了基本典型李超代数的普遍包络代数的中心结构.设g=g0(?)g1=Lie(G)是基本典型李超代数,其中G是经典的代数超群,其纯偶群Gev是简约群,满足Lie(Gev)=g0.首先我们证明了中心是交换环,接着本文用两种办法证明了Z(g)无U(g)的零因子,然后说明了包络代数经过基域变换后,作为该中心的分式域上的代数是中心单代数.从而,解决了舒斌-郑立笋在基本典型李超代数包络代数中心结构的研究中提出的相关猜想.于是完整给出了该中心的结构,即包络的中心Z(g)的分式域等于由p-中心以及中心中Gev-不变部分Z(g)Gev所生成的子代数的分式域.
第二部分我们主要研究了g=osp(1|2n)的表示.我们首先证明了此时的包络代数的中心是整闭的,再结合第一部分的结论,得到了Z(g)恰好由P-中心以及中心中Gev-不变部分生成.在这个证明过程中,我们还得到了代数Z(g)Gev与u(η)W间的同构,这也推广了李代数情形的Harish-Chandra定理.接着我们利用范畴等价的方法以及反中心的性质,具体给出了当p-特征函数x∈g*是正则半单和正则幂零时的Ux(g)的偶模范畴的块分解.特别地,我们得到了X正则幂零时,Ux(g)的不可约模的同构类的分类；并且利用投射维数的性质,证明了此时的baby Verma模都不是投射模,这个结论推广了王伟强-赵磊在osp(1|2)情形的结果.接着,我们给出了Z(U(osp(1|2)))的极大谱的光滑点和ospp(1|2)的不可约模之间的关系.这个结果和典型李代数的情形不一样,因为此时存在一个不是最大维数的不可约模,它的中心零化子却是光滑点.最后,我们完全给出了osp(1|2)的所有偶模范畴的块分解.
第三部分我们研究了李代数的幂零锥和它的不变多项式环A.A.Premet在1990年猜想：所有有限维限制李代数的幂零锥都是不可约的.对于典型李代数,该猜想所陈述的是已广为认知的结论.对于Jacoboson-Witt代数,Premet本人在文献[63]中给予了证明,并且在同一文献中他还证明了该代数的包络代数在Jacobson-Witt代数自同构群作用下的不变量环可由类似于经典情形Chevalley限制定理的结果刻画.本文的研究对象是S型的Cartan型李代数.我们通过构造幂零锥的一个稠密子集证明了这个猜想对S型也是成立的.同时,我们证明了特殊代数Sn的幂零锥是个完全交.对于它的不变多项式环,我们在[9]的基础上证明了下面的结论：k[Sn]Aut(sn)(?)k[T'nZ]Weyl Group,其中Tn′是Sn的广义环面.这个结论改进了[9]的相关结果也是典型李代数的Chevally限制定理的推广.作为推论,我们得到了Sn有无穷多个幂零轨道,这也部分证明了姚裕丰和舒斌的一个猜想.
This thesis is mainly on the study of basic classical Lie superalgebra and Cartan type Lie algebra over an algebraically closed field k of characteristic p>2and their representations. The main results are listed below:
The first main result is on the center of the universal enveloping algebra of a basic classical Lie superalgebra. Let g=g0+g1be such a Lie superalgebra over k. There is an algebraic supergroup G satisfying Lie(G)=g, with the purely even subgroup Gev which is a connected reductive group. The center Z(g) of the universal enveloping algebra U(g) turns out to be commutative. Then we prove in two different ways that the center Z(g) is an integral domain, and furthermore prove that U(g)(?)z(g) Frac(Z(g)) is a simple superalgebra over Frac(Z(g)). This enables us to prove the conjecture proposed by Bin Shu and Lisun Zheng about the structure of the center of enveloping algebra. That is, the quotient field of Z(g) coincides with that of the subalgebra generated by the Gev-invariant ring Z(g)Gev of Z(g) and the p-center Zo of U(g0).
The second main result is on the representations of Lie superalgebra g=osp(1|2n). As an application of the first main result to this special case, we first show that Z(U(g)) is just generated by Z(g)Gev and Z0. We can also get an algebraic isomorphism between Z(g)Gev and U(η))w, which generalizes the Harish-Chandra theorem in the case of complex semi-simple Lie algebras. Then we investigate the blocks of the underlying even category of Ux(g) for χ∈g*. By equivalence of categories and the property of the anti-center of U(g), we de-scribe precisely the blocks when χ is regular semisimple and regular nilpotent. In particular, when χ is regular nilpotent, we classify the isomorphism classes of irreducible modules; moreover, we show that in this case there is no projective baby Verma module by computing dimensions, generalizing the result of Wang-Zhao for osp(1|2). Next, we give the precise relation between the smooth points of the maximal spectrum Maxspec (Z(g)) and the corresponding irreducible mod- ules for osp(1|2). This result is different from the case of classical Lie algebra. Finally, we count all blocks in the case for osp(1|2).
The third result is on the irreducibility of the variety of nilpotent elements in a Lie algebra and the ring of invariant polynomial functions on the Lie algebra. A.A.Premet conjectured that this variety is irreducible for any finite dimensional restricted Lie algebra. It is well known that this variety is irreducible for any classical Lie algebra. And then, Premet in [63] confirmed this conjecture for Jacoboson-Witt algebra. In the same paper, he described the ring of invariant polynomial functions on the Jacoboson-Witt algebra by a Chevalley restricted theorem similar to the classical Lie algebras. In the present thesis, we study the the Lie algebra Sn of Cartan type of S type. The conjecture for type S is showed to be true by constructing a dense subset. Moreover, It can be shown that the variety of nilpotent elements in the algebra Sn is a complete intersection. Motivated by the proof of the irreducibility, we show that k[Sn]Aut(Sn)(?)k[T'n]WeylGroup, where T'n is a generic torus in Sn. This result also generalizes the Chevalley restrict theorem of classical Lie algebras. As a byproduct, we also prove that there are infinitely many nilpotent orbits in Sn, which give a positive answer to a conjecture of Yufeng Yao and Bin Shu.
第一部分刻画了基本典型李超代数的普遍包络代数的中心结构.设g=g0(?)g1=Lie(G)是基本典型李超代数,其中G是经典的代数超群,其纯偶群Gev是简约群,满足Lie(Gev)=g0.首先我们证明了中心是交换环,接着本文用两种办法证明了Z(g)无U(g)的零因子,然后说明了包络代数经过基域变换后,作为该中心的分式域上的代数是中心单代数.从而,解决了舒斌-郑立笋在基本典型李超代数包络代数中心结构的研究中提出的相关猜想.于是完整给出了该中心的结构,即包络的中心Z(g)的分式域等于由p-中心以及中心中Gev-不变部分Z(g)Gev所生成的子代数的分式域.
第二部分我们主要研究了g=osp(1|2n)的表示.我们首先证明了此时的包络代数的中心是整闭的,再结合第一部分的结论,得到了Z(g)恰好由P-中心以及中心中Gev-不变部分生成.在这个证明过程中,我们还得到了代数Z(g)Gev与u(η)W间的同构,这也推广了李代数情形的Harish-Chandra定理.接着我们利用范畴等价的方法以及反中心的性质,具体给出了当p-特征函数x∈g*是正则半单和正则幂零时的Ux(g)的偶模范畴的块分解.特别地,我们得到了X正则幂零时,Ux(g)的不可约模的同构类的分类；并且利用投射维数的性质,证明了此时的baby Verma模都不是投射模,这个结论推广了王伟强-赵磊在osp(1|2)情形的结果.接着,我们给出了Z(U(osp(1|2)))的极大谱的光滑点和ospp(1|2)的不可约模之间的关系.这个结果和典型李代数的情形不一样,因为此时存在一个不是最大维数的不可约模,它的中心零化子却是光滑点.最后,我们完全给出了osp(1|2)的所有偶模范畴的块分解.
第三部分我们研究了李代数的幂零锥和它的不变多项式环A.A.Premet在1990年猜想：所有有限维限制李代数的幂零锥都是不可约的.对于典型李代数,该猜想所陈述的是已广为认知的结论.对于Jacoboson-Witt代数,Premet本人在文献[63]中给予了证明,并且在同一文献中他还证明了该代数的包络代数在Jacobson-Witt代数自同构群作用下的不变量环可由类似于经典情形Chevalley限制定理的结果刻画.本文的研究对象是S型的Cartan型李代数.我们通过构造幂零锥的一个稠密子集证明了这个猜想对S型也是成立的.同时,我们证明了特殊代数Sn的幂零锥是个完全交.对于它的不变多项式环,我们在[9]的基础上证明了下面的结论：k[Sn]Aut(sn)(?)k[T'nZ]Weyl Group,其中Tn′是Sn的广义环面.这个结论改进了[9]的相关结果也是典型李代数的Chevally限制定理的推广.作为推论,我们得到了Sn有无穷多个幂零轨道,这也部分证明了姚裕丰和舒斌的一个猜想.
This thesis is mainly on the study of basic classical Lie superalgebra and Cartan type Lie algebra over an algebraically closed field k of characteristic p>2and their representations. The main results are listed below:
The first main result is on the center of the universal enveloping algebra of a basic classical Lie superalgebra. Let g=g0+g1be such a Lie superalgebra over k. There is an algebraic supergroup G satisfying Lie(G)=g, with the purely even subgroup Gev which is a connected reductive group. The center Z(g) of the universal enveloping algebra U(g) turns out to be commutative. Then we prove in two different ways that the center Z(g) is an integral domain, and furthermore prove that U(g)(?)z(g) Frac(Z(g)) is a simple superalgebra over Frac(Z(g)). This enables us to prove the conjecture proposed by Bin Shu and Lisun Zheng about the structure of the center of enveloping algebra. That is, the quotient field of Z(g) coincides with that of the subalgebra generated by the Gev-invariant ring Z(g)Gev of Z(g) and the p-center Zo of U(g0).
The second main result is on the representations of Lie superalgebra g=osp(1|2n). As an application of the first main result to this special case, we first show that Z(U(g)) is just generated by Z(g)Gev and Z0. We can also get an algebraic isomorphism between Z(g)Gev and U(η))w, which generalizes the Harish-Chandra theorem in the case of complex semi-simple Lie algebras. Then we investigate the blocks of the underlying even category of Ux(g) for χ∈g*. By equivalence of categories and the property of the anti-center of U(g), we de-scribe precisely the blocks when χ is regular semisimple and regular nilpotent. In particular, when χ is regular nilpotent, we classify the isomorphism classes of irreducible modules; moreover, we show that in this case there is no projective baby Verma module by computing dimensions, generalizing the result of Wang-Zhao for osp(1|2). Next, we give the precise relation between the smooth points of the maximal spectrum Maxspec (Z(g)) and the corresponding irreducible mod- ules for osp(1|2). This result is different from the case of classical Lie algebra. Finally, we count all blocks in the case for osp(1|2).
The third result is on the irreducibility of the variety of nilpotent elements in a Lie algebra and the ring of invariant polynomial functions on the Lie algebra. A.A.Premet conjectured that this variety is irreducible for any finite dimensional restricted Lie algebra. It is well known that this variety is irreducible for any classical Lie algebra. And then, Premet in [63] confirmed this conjecture for Jacoboson-Witt algebra. In the same paper, he described the ring of invariant polynomial functions on the Jacoboson-Witt algebra by a Chevalley restricted theorem similar to the classical Lie algebras. In the present thesis, we study the the Lie algebra Sn of Cartan type of S type. The conjecture for type S is showed to be true by constructing a dense subset. Moreover, It can be shown that the variety of nilpotent elements in the algebra Sn is a complete intersection. Motivated by the proof of the irreducibility, we show that k[Sn]Aut(Sn)(?)k[T'n]WeylGroup, where T'n is a generic torus in Sn. This result also generalizes the Chevalley restrict theorem of classical Lie algebras. As a byproduct, we also prove that there are infinitely many nilpotent orbits in Sn, which give a positive answer to a conjecture of Yufeng Yao and Bin Shu.
引文
[1]D. Arnaudon, M. Bauer, and L. Frappatm, On Casimir's Ghost, Com-mun.Math.Phys.187,pp.429-439,1997.
[2]A. A. Albert and M. S.Frank, Simple Lie algebras of characteristic p, Univ. Politec Torino Rend. Sem. Mat.14, pp.117-139,1954.
[3]M. Aubry and T. M. Lemaire, Zero divisors in enveloping algebras of graded Lie algebras, J. Pure Appl. Algebra 38, pp.159-166,1985.
[4]M. F. Atyah, I. G. McDonald, Introduction to commutative algebra, Addison-Wesley Publishing company,1969.
[5]F. A. Berezin, Introduction to superanalysis, D. Reidel Publishing Company Dordrecht.
[6]H. Boseck, Classical Lie supergroups, Math. Nachr.148, pp.81-115,1990.
[7]E. J. Behr, Enveloping algebras of Lie superalgebras, Pacific J. Math.130, pp.9-25,1987.
[8]A. D. Bell, A criterion for primeness of enveloping algebras of Lie superal-gebras, J.Pure and Applied Algebra 69, pp.111-120,1990.
[9]J. M. Bois and R. Farnsteiner and B. Shu, Weyl groups for non-classical restricted Lie algebras and the Chevalley Restriction Theorem, Forum Mathematicum, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: 10.1515/forum-2011-0145,2012.
[10]K. A. Brown and C. R. Hajarnavis, Injectively homogeneous rings, J. Pure Appl. Algebra 51, pp.65-77,1988.
[11]J. Brundan and A. Kleshchev, Projective representations of symmetric groups via Sergeev duality, Math. Z.239,pp.27-68,2002.
[12]J. Brundan and A. Kleshchev, Modular representations of the supergroup Q(n), I, J. Algebra 260, PP.64-98,2003.
[13]J. Brundan and J. Kujawa, A new proof of the Mullineux conjecture, J. Alg. Combinatorics 18, pp.13-39,2003.
[14]K. A. Brown, I. Gordon, The ramification of centres:Lie algebras in positive characteristic and quantised enveloping algebras, Math. Z.238,pp.733-779, 2001.
[15]K. A. Brown, K. R. Goodearl, Homological Aspects of Noetherian PI Hopf Algebras and Irreducible Modules of Maximal Dimension, Journal of algebra 198, pp.240-265,1997.
[16]Y. Bahturin and M. Kochetov, Group gradings on restricted Cartan type Lie algebra, Pacific Journal of Mathematics,253, no.2, p.289-319,2011.
[17]R. Bovad, Some elementary results on the cohomology of graded Lie algebra, Asterisque 113-114, pp.156-166, Soc. Math. France, Paris,1984.
[18]R. E. Block and R. L. Wilson, The restricted simple Lie algebras are of classical or Cartan type, Proc. Natl. Acad. Sci. USA, Vol.81, pp.5271-5274, 1984.
[19]Bin Shu and W. Wang, Modular representations of the Otho- Syplectic su-pergroups, Proc. London Math. Soc. (3) 96, pp.251-271,2008.
[20]H. J. Chang, Uber Wittsche Lie-Ringe, Abh. Math. Semin. Uhiv. Hamb.14, pp.151-184,1941.
[21]H. Cartan and S. Eilenberg, Homological Algebra, Princeton Univ. Press, Princeton,1956.
[22]S.-J. Cheng and N. Lam, Irreducible characters of general linear superalgebra and super duality, Communications in Mathematical Physics,298, no.3, pp. 645-672,2010.
[23]S.-J. Cheng, N. Lam and W.Wang, Super duality and irreducible characters of ortho-symplectic Lie superalgebras, Invent.Math.183, pp.189-224,2011.
[24]S.-J. Cheng, W.Wang, Dualities and Representations of Lie Superalgebras, GSM 144, Amer. Math. Soc., Providence, RI,2013.
[25]S. P. Demushkin, Cartan subalgebras of simple Lie p-algebras Wn and Sn. Sibirskii Matematicheskii Zhurnal, Vol.11, No.2, pp.310-325,1970.
[26]D. Eisenbud, Subrings of artinian and noetherian rings, Math. Ann.185, pp. 247-249,1970.
[27]M. S. Frank, A new class of simple Lie algebras, Proc. Natl. Acad. Sci. USA 40, pp.713-719,1954.
[28]M. S. Frank, Two new classes of simple Lie algebras, Trans. Am. Math. Soc. 112,456-482,1964.
[29]R. Fioresi and F. Gavarini, Chevalley supergroups, Mem. Amer. Math. Soc. 215, no.1014,2012.
[30]R. Fioresi and F. Gavarini, Algebraic supergroups with Lie superalgebras of classical type, Journal of Lie Theory 23, pp.143-158,2013.
[31]M. K. Gaillard, B.Zumino, Supersymmetry and superstring phenomenology, Eru. Phys. J.C 59, pp.213-221,2009.
[32]M. A. Golberg, The Derivative of a Determinant The American Mathemat-ical Monthly Vol.79, No.10, pp.1124-112,1972
[33]M. Gorelik, On the ghost centre of Lie superalgebras, Annales de Linstitut Fourier 50, pp.1745-1764,2000.
[34]K. R. Goodearl, R. B. Warfield Jr, An Introduction to Noncommutative Noetherian Rings, Number 16 in London Mathematical Society Student Texts. Cambridge University Press,1989
[35]J. E. Humphreys, Linear Algebraic Groups, Graduate Texts in Mathematics 21, Springer-Verlag, New York etc,1975.
[36]J. E. Humphreys, Modular representations of simple Lie algebras, Bull. Amer. Math. Soc.35, (2) pp.105-122,1998.
[37]R. Hartshorne, Algebraic Geometry. Graduate Texts in Math., vol.52, Springer-Verlag, Berlin and New York,1977.
[38]E. Hille, Lectures on Ordinary Differential Equations, Addison-Wesley, Read-ing, Mass.,1969.
[39]D. J. Hemmer, J. Kujawa and D.K. Nakano, Representation type of Schur superalgebras J. Group Theory 9, pp.283-306,2006.
[40]N. Jacobson, Classes of restricted Lie algebras of characteristic p II, Duke Math. J.10,pp.107-121,1943.
[41]J. C. Jantzen, Representations of Lie algebras in prime characteristic. In A. Broer, editor, Representation Theories and Algebraic Geometry, Proceedings Montr'eal 1997 (NATO ASI Series C 514), pp.185-235. Dordrecht etc, Kluwer,1998
[42]J. C. Jantzen, Nilpotent orbits in representation theory, pages 1-221, in "Lie Theory, Lie algebras and representations", Progress in Mathematics 228, Birhauser, Boston.Basel.Berlin,2004.
[43]T. Jozefiak, Semisimple superalgebras, Lecture Notes in Math.1352, pp.96-113, Springer, Berlin,1988.
[44]V. G. Kac, Representations of classical Lie superalgebras, Lecture Notes in Math,676, pp.597-626,1978.
[45]V. G. Kac, Classification of simple Lie superalgebras, Functional Anal. Appl, pp.263-265,1975
[46]V. G. Kac, Lie superalgebras, Adv. Math.26, pp.8-96,1977.
[47]V. G. Kac, Characters of typical representations of classical lie superalgebras, Commun Algebra, vol.5, no.8, pp.889-897,1977.
[48]B. Kostant, Lie group representations on polynomial rings. Amer. J. Math., 85, pp.327-404,1963.
[49]A. I. Kostrikin and I. R. Safarevic, Cartan pseudogroups and Lie p-algebras, Dokl. Akad.Nauk SSSR 168, pp.740-742,1966.
[50]J. Kujawa, The Steinberg tensor product theorem for GL(m|n), Represen-tations of algebraic groups, Quantum groups and Lie algebras, Contemp. Math.413,2006.
[51]V. Kac and B. Weisfeiler, Coadjoint action of a semi-simple algebraic group and the center of the enveloping algebra in characteristic p, Indag. Math.38, pp.136-151,1976.
[52]T. Y. Lam, A first course in noncommutative rings, Graduate Texts in Math-ematics,131 (2nd ed.), Berlin, New York:Springer-Verlag, ISBN 978-0-387-95183-6, MR 1838439,2001.
[53]D. Leites, Introduction to the theory of supermanifold, Russian Math. surveys 35, pp.1-64,1980.
[54]Z. Lin and D. K. Nakano, Algebraic group actions in the cohomology theory of Lie algebra of Cartan type, Journal of Algebra,179, pp.852-888,1996.
[55]Y. I. Manin, Gauge Field Theory and Complex Geometry, Grundlehren der mathematischen Wissenschaften 289, second edition, Spriger- Verlag,1997.
[56]J. Milnor and J. Moore, On the structure of Hopf algebras, Ann. Math.81, pp.211-264,1965.
[57]J. C. McConnell and J. C. Robson, Noncommutative Noetherian Rings, Wiley-Interscience, New York,1987.
[58]I. Mirkovic and D. Rumynin, Centers of reducecd enveloping algebra, Maht. Z.231,123-132,1999.
[59]W. Mills and G. B. Seligman, Lie algebras of classical type, J. Math. Mech. 6, pp.519-548,1957.
[60]B. J. Muller, Localization in non-commutative Noetherian rings, Canad. J. Math.28, pp.600-610,1976
[61]I. M. Musson, On the Center of the Enveloping Algebra of a Classical Simple Lie Superalgebra, J. of Algebra 193 285-308,1997.
[62]A. A. Premet, Regular Cartan subalgebras and nilpotent elements in restricted Lie algebras. Math. USSR. Sbornik, Vol.66, No.6, pp.555-570,1990.
[63]A. A. Premet, The theorem on restriction of invariants, and nilpotent ele-ments in Wn. Math. USSR. Sbornik, Vol.73, No.1, pp.135-159,1992.
[64]A. A. Premet, Nilpotent commuting varieties of reductive Lie algebra. Invent. math., Vol.154, No.3, pp.653-683,2003.
[65]J. J. Rotman, An Introduction to Homological Algebra, Academic Press, 1979.
[66]S. Skryabin, Invariant polynomial functions on the Poisson algebra in char-acteristic p, J.Algebra 256, pp.146-179,2002.
[67]R. Shafarevich, Basic algebraic geometry, "Nauka", Moscow,1972; English transl. Springer Verlag, Berlin,1974.
[68]V. Serganova, Kazhdan-Lusztig polynomials and character formula for the Lie superalgebra gl(m,n), Selecta Math.2, pp.607-654,1996.
[69]A. Sergeev, The invariant polynomials on simple Lie superalgebras, An Elec. J. AMS, Vol 3, PP.250-280,1999.
[70]M. Scheunet, The theroy of Lie superalgebras, Lecture Notes in Mathematics 716,1979.
[71]H. Strade and R. Farnsteiner, Modular Lie algebras and their representations, Marcel Dekker, New York,1988.
[72]B. Shu and W. Wang, Modular representations of the otho-syplectic super-groups, Proc. London Math. Soc.96, pp.251-271,2008.
[73]B. Shu and L.S. Zheng, On Lie superalgebras of algebraic supergroups, Alge-bra Colloquium 16, pp.361-370,2009.
[74]B. Shu, C. W. Zhang, Representations of the restricted Cartan type Lie superalgebras, preprint,2008.
[75]J. T. Stafford and J. J. Zhang, Homological properties of (graded) Noetherian PI rings, Journal of Algebra 168, pp.988-1026,1994.
[76]V. S. Varadarjan, Supersymmetry for mathematicians; an introduction, Courant Lecture.
[77]F. D. Veldkamp, The center of the universal enveloping algebra of a Lie algebra in characteristic p, Ann. Sci. Ecole Norm. Sup.5,217-240,1972.
[78]W. C. Waterhouse, Automorphism schemes and forms of Witt Lie algebras, J. Algebra,17 pp.34.40,1971.
[79]J. Wei, H. Chang and X. Lu, The variety of nilpotent elements and invariant polynomial functions on the special algebra Sn.Accepted by Forum mathe-maticum.
[80]R. Wilson, Automorphisms of graded Lie algebra of Cartan type. Comm. Algebra,3(7), pp.591-613,1975.
[81]W. Wang and L. Zhao, Representations of Lie superalgebras in prime char-acteristic I, Proc. London Math. Soc.99, pp.145-167,2009. Notes 11, Amer. Math. Soc., Providence, RI,2004.
[82]Y. F. Yao and B. Shu, Nilpotent Orbits in the Witt Algebra W1. Comm. Algebra,39(9), pp.3232-3241,2011.
[83]Y. F. Yao, Note on primitive ideals of enveloping algebras in prime charac-teristic, Algebra Colloquium 18:4, pp.701-708,2011.
[84]H. Zassenhaus, Uber Lie'sche Ringe mit Prirnzahl Characteristik, Abh. Math. Semin. Univ. Hamb.13, pp.1-100,1939.
[85]H. Zassenhaus, The representations of Lie algebras of prime characteristec, Proc. Glasgow Math. Ass.21, pp.1-36,1954.
[86]Y. Zhong, Injective homogeneity and the Auslander-Gorenstein property, Glasgow Math. J.37, pp.191-204,1995.
[87]L. Zhao, Representations of Lie superalgebras in prime characteristic Ⅲ, arxiv:0910.20,2009.
L. S. Zheng, PhD thesis in Chinese, ECNU,2009.
[2]A. A. Albert and M. S.Frank, Simple Lie algebras of characteristic p, Univ. Politec Torino Rend. Sem. Mat.14, pp.117-139,1954.
[3]M. Aubry and T. M. Lemaire, Zero divisors in enveloping algebras of graded Lie algebras, J. Pure Appl. Algebra 38, pp.159-166,1985.
[4]M. F. Atyah, I. G. McDonald, Introduction to commutative algebra, Addison-Wesley Publishing company,1969.
[5]F. A. Berezin, Introduction to superanalysis, D. Reidel Publishing Company Dordrecht.
[6]H. Boseck, Classical Lie supergroups, Math. Nachr.148, pp.81-115,1990.
[7]E. J. Behr, Enveloping algebras of Lie superalgebras, Pacific J. Math.130, pp.9-25,1987.
[8]A. D. Bell, A criterion for primeness of enveloping algebras of Lie superal-gebras, J.Pure and Applied Algebra 69, pp.111-120,1990.
[9]J. M. Bois and R. Farnsteiner and B. Shu, Weyl groups for non-classical restricted Lie algebras and the Chevalley Restriction Theorem, Forum Mathematicum, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: 10.1515/forum-2011-0145,2012.
[10]K. A. Brown and C. R. Hajarnavis, Injectively homogeneous rings, J. Pure Appl. Algebra 51, pp.65-77,1988.
[11]J. Brundan and A. Kleshchev, Projective representations of symmetric groups via Sergeev duality, Math. Z.239,pp.27-68,2002.
[12]J. Brundan and A. Kleshchev, Modular representations of the supergroup Q(n), I, J. Algebra 260, PP.64-98,2003.
[13]J. Brundan and J. Kujawa, A new proof of the Mullineux conjecture, J. Alg. Combinatorics 18, pp.13-39,2003.
[14]K. A. Brown, I. Gordon, The ramification of centres:Lie algebras in positive characteristic and quantised enveloping algebras, Math. Z.238,pp.733-779, 2001.
[15]K. A. Brown, K. R. Goodearl, Homological Aspects of Noetherian PI Hopf Algebras and Irreducible Modules of Maximal Dimension, Journal of algebra 198, pp.240-265,1997.
[16]Y. Bahturin and M. Kochetov, Group gradings on restricted Cartan type Lie algebra, Pacific Journal of Mathematics,253, no.2, p.289-319,2011.
[17]R. Bovad, Some elementary results on the cohomology of graded Lie algebra, Asterisque 113-114, pp.156-166, Soc. Math. France, Paris,1984.
[18]R. E. Block and R. L. Wilson, The restricted simple Lie algebras are of classical or Cartan type, Proc. Natl. Acad. Sci. USA, Vol.81, pp.5271-5274, 1984.
[19]Bin Shu and W. Wang, Modular representations of the Otho- Syplectic su-pergroups, Proc. London Math. Soc. (3) 96, pp.251-271,2008.
[20]H. J. Chang, Uber Wittsche Lie-Ringe, Abh. Math. Semin. Uhiv. Hamb.14, pp.151-184,1941.
[21]H. Cartan and S. Eilenberg, Homological Algebra, Princeton Univ. Press, Princeton,1956.
[22]S.-J. Cheng and N. Lam, Irreducible characters of general linear superalgebra and super duality, Communications in Mathematical Physics,298, no.3, pp. 645-672,2010.
[23]S.-J. Cheng, N. Lam and W.Wang, Super duality and irreducible characters of ortho-symplectic Lie superalgebras, Invent.Math.183, pp.189-224,2011.
[24]S.-J. Cheng, W.Wang, Dualities and Representations of Lie Superalgebras, GSM 144, Amer. Math. Soc., Providence, RI,2013.
[25]S. P. Demushkin, Cartan subalgebras of simple Lie p-algebras Wn and Sn. Sibirskii Matematicheskii Zhurnal, Vol.11, No.2, pp.310-325,1970.
[26]D. Eisenbud, Subrings of artinian and noetherian rings, Math. Ann.185, pp. 247-249,1970.
[27]M. S. Frank, A new class of simple Lie algebras, Proc. Natl. Acad. Sci. USA 40, pp.713-719,1954.
[28]M. S. Frank, Two new classes of simple Lie algebras, Trans. Am. Math. Soc. 112,456-482,1964.
[29]R. Fioresi and F. Gavarini, Chevalley supergroups, Mem. Amer. Math. Soc. 215, no.1014,2012.
[30]R. Fioresi and F. Gavarini, Algebraic supergroups with Lie superalgebras of classical type, Journal of Lie Theory 23, pp.143-158,2013.
[31]M. K. Gaillard, B.Zumino, Supersymmetry and superstring phenomenology, Eru. Phys. J.C 59, pp.213-221,2009.
[32]M. A. Golberg, The Derivative of a Determinant The American Mathemat-ical Monthly Vol.79, No.10, pp.1124-112,1972
[33]M. Gorelik, On the ghost centre of Lie superalgebras, Annales de Linstitut Fourier 50, pp.1745-1764,2000.
[34]K. R. Goodearl, R. B. Warfield Jr, An Introduction to Noncommutative Noetherian Rings, Number 16 in London Mathematical Society Student Texts. Cambridge University Press,1989
[35]J. E. Humphreys, Linear Algebraic Groups, Graduate Texts in Mathematics 21, Springer-Verlag, New York etc,1975.
[36]J. E. Humphreys, Modular representations of simple Lie algebras, Bull. Amer. Math. Soc.35, (2) pp.105-122,1998.
[37]R. Hartshorne, Algebraic Geometry. Graduate Texts in Math., vol.52, Springer-Verlag, Berlin and New York,1977.
[38]E. Hille, Lectures on Ordinary Differential Equations, Addison-Wesley, Read-ing, Mass.,1969.
[39]D. J. Hemmer, J. Kujawa and D.K. Nakano, Representation type of Schur superalgebras J. Group Theory 9, pp.283-306,2006.
[40]N. Jacobson, Classes of restricted Lie algebras of characteristic p II, Duke Math. J.10,pp.107-121,1943.
[41]J. C. Jantzen, Representations of Lie algebras in prime characteristic. In A. Broer, editor, Representation Theories and Algebraic Geometry, Proceedings Montr'eal 1997 (NATO ASI Series C 514), pp.185-235. Dordrecht etc, Kluwer,1998
[42]J. C. Jantzen, Nilpotent orbits in representation theory, pages 1-221, in "Lie Theory, Lie algebras and representations", Progress in Mathematics 228, Birhauser, Boston.Basel.Berlin,2004.
[43]T. Jozefiak, Semisimple superalgebras, Lecture Notes in Math.1352, pp.96-113, Springer, Berlin,1988.
[44]V. G. Kac, Representations of classical Lie superalgebras, Lecture Notes in Math,676, pp.597-626,1978.
[45]V. G. Kac, Classification of simple Lie superalgebras, Functional Anal. Appl, pp.263-265,1975
[46]V. G. Kac, Lie superalgebras, Adv. Math.26, pp.8-96,1977.
[47]V. G. Kac, Characters of typical representations of classical lie superalgebras, Commun Algebra, vol.5, no.8, pp.889-897,1977.
[48]B. Kostant, Lie group representations on polynomial rings. Amer. J. Math., 85, pp.327-404,1963.
[49]A. I. Kostrikin and I. R. Safarevic, Cartan pseudogroups and Lie p-algebras, Dokl. Akad.Nauk SSSR 168, pp.740-742,1966.
[50]J. Kujawa, The Steinberg tensor product theorem for GL(m|n), Represen-tations of algebraic groups, Quantum groups and Lie algebras, Contemp. Math.413,2006.
[51]V. Kac and B. Weisfeiler, Coadjoint action of a semi-simple algebraic group and the center of the enveloping algebra in characteristic p, Indag. Math.38, pp.136-151,1976.
[52]T. Y. Lam, A first course in noncommutative rings, Graduate Texts in Math-ematics,131 (2nd ed.), Berlin, New York:Springer-Verlag, ISBN 978-0-387-95183-6, MR 1838439,2001.
[53]D. Leites, Introduction to the theory of supermanifold, Russian Math. surveys 35, pp.1-64,1980.
[54]Z. Lin and D. K. Nakano, Algebraic group actions in the cohomology theory of Lie algebra of Cartan type, Journal of Algebra,179, pp.852-888,1996.
[55]Y. I. Manin, Gauge Field Theory and Complex Geometry, Grundlehren der mathematischen Wissenschaften 289, second edition, Spriger- Verlag,1997.
[56]J. Milnor and J. Moore, On the structure of Hopf algebras, Ann. Math.81, pp.211-264,1965.
[57]J. C. McConnell and J. C. Robson, Noncommutative Noetherian Rings, Wiley-Interscience, New York,1987.
[58]I. Mirkovic and D. Rumynin, Centers of reducecd enveloping algebra, Maht. Z.231,123-132,1999.
[59]W. Mills and G. B. Seligman, Lie algebras of classical type, J. Math. Mech. 6, pp.519-548,1957.
[60]B. J. Muller, Localization in non-commutative Noetherian rings, Canad. J. Math.28, pp.600-610,1976
[61]I. M. Musson, On the Center of the Enveloping Algebra of a Classical Simple Lie Superalgebra, J. of Algebra 193 285-308,1997.
[62]A. A. Premet, Regular Cartan subalgebras and nilpotent elements in restricted Lie algebras. Math. USSR. Sbornik, Vol.66, No.6, pp.555-570,1990.
[63]A. A. Premet, The theorem on restriction of invariants, and nilpotent ele-ments in Wn. Math. USSR. Sbornik, Vol.73, No.1, pp.135-159,1992.
[64]A. A. Premet, Nilpotent commuting varieties of reductive Lie algebra. Invent. math., Vol.154, No.3, pp.653-683,2003.
[65]J. J. Rotman, An Introduction to Homological Algebra, Academic Press, 1979.
[66]S. Skryabin, Invariant polynomial functions on the Poisson algebra in char-acteristic p, J.Algebra 256, pp.146-179,2002.
[67]R. Shafarevich, Basic algebraic geometry, "Nauka", Moscow,1972; English transl. Springer Verlag, Berlin,1974.
[68]V. Serganova, Kazhdan-Lusztig polynomials and character formula for the Lie superalgebra gl(m,n), Selecta Math.2, pp.607-654,1996.
[69]A. Sergeev, The invariant polynomials on simple Lie superalgebras, An Elec. J. AMS, Vol 3, PP.250-280,1999.
[70]M. Scheunet, The theroy of Lie superalgebras, Lecture Notes in Mathematics 716,1979.
[71]H. Strade and R. Farnsteiner, Modular Lie algebras and their representations, Marcel Dekker, New York,1988.
[72]B. Shu and W. Wang, Modular representations of the otho-syplectic super-groups, Proc. London Math. Soc.96, pp.251-271,2008.
[73]B. Shu and L.S. Zheng, On Lie superalgebras of algebraic supergroups, Alge-bra Colloquium 16, pp.361-370,2009.
[74]B. Shu, C. W. Zhang, Representations of the restricted Cartan type Lie superalgebras, preprint,2008.
[75]J. T. Stafford and J. J. Zhang, Homological properties of (graded) Noetherian PI rings, Journal of Algebra 168, pp.988-1026,1994.
[76]V. S. Varadarjan, Supersymmetry for mathematicians; an introduction, Courant Lecture.
[77]F. D. Veldkamp, The center of the universal enveloping algebra of a Lie algebra in characteristic p, Ann. Sci. Ecole Norm. Sup.5,217-240,1972.
[78]W. C. Waterhouse, Automorphism schemes and forms of Witt Lie algebras, J. Algebra,17 pp.34.40,1971.
[79]J. Wei, H. Chang and X. Lu, The variety of nilpotent elements and invariant polynomial functions on the special algebra Sn.Accepted by Forum mathe-maticum.
[80]R. Wilson, Automorphisms of graded Lie algebra of Cartan type. Comm. Algebra,3(7), pp.591-613,1975.
[81]W. Wang and L. Zhao, Representations of Lie superalgebras in prime char-acteristic I, Proc. London Math. Soc.99, pp.145-167,2009. Notes 11, Amer. Math. Soc., Providence, RI,2004.
[82]Y. F. Yao and B. Shu, Nilpotent Orbits in the Witt Algebra W1. Comm. Algebra,39(9), pp.3232-3241,2011.
[83]Y. F. Yao, Note on primitive ideals of enveloping algebras in prime charac-teristic, Algebra Colloquium 18:4, pp.701-708,2011.
[84]H. Zassenhaus, Uber Lie'sche Ringe mit Prirnzahl Characteristik, Abh. Math. Semin. Univ. Hamb.13, pp.1-100,1939.
[85]H. Zassenhaus, The representations of Lie algebras of prime characteristec, Proc. Glasgow Math. Ass.21, pp.1-36,1954.
[86]Y. Zhong, Injective homogeneity and the Auslander-Gorenstein property, Glasgow Math. J.37, pp.191-204,1995.
[87]L. Zhao, Representations of Lie superalgebras in prime characteristic Ⅲ, arxiv:0910.20,2009.
L. S. Zheng, PhD thesis in Chinese, ECNU,2009.