Jacobson-Witt代数的不可约表示
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摘要
本文利用一般线性群的多项式表示分解理论,来讨论了Jacobson-Witt代数的不可约表示。在Jacobson-Witt代数W(n)的阶化结构下,证明了它的次数小于等于p-n的阶化子空间可以做不可约GL(n)的多项式表示分解。并主要研究了具有高度小于等于p+1-n的p特征函数χ,当χ的秩等于1时,W(3)的不可约模的维数以及同构类个数问题。
In this paper, we mainly used polynomial representations decomposition theory of general linear group to concern irreducible representations of Jacobson-Witt algebra. Under the graded construction of Jacobson-Witt algebra , we have proved that it's graded subspace of degree less than p - n can be decomposed into two irreducible GL(n) modules . we mainly researched dimensions and classes of irreducible modules of W(3) with p-character x and rank(x) equal to 1.
In this paper, we mainly used polynomial representations decomposition theory of general linear group to concern irreducible representations of Jacobson-Witt algebra. Under the graded construction of Jacobson-Witt algebra , we have proved that it's graded subspace of degree less than p - n can be decomposed into two irreducible GL(n) modules . we mainly researched dimensions and classes of irreducible modules of W(3) with p-character x and rank(x) equal to 1.
引文
[1] Chang Ho-Jui, Uber Wittsche Lie-Ringe, Abh. Math Sem. Univ. Hamburg 14(1941), 151-184.
[2] Benkart G.M, Cartan subalgebras in Lie algebras of cartan type, Lie Algebras and Related Topics, CMS Conf. proc 5 (1986), 157-187.
[3] Shen Guang-Yu, Graded modules of graded Lie algebras of Cartan type, I ,Scientica Sinica 29(1986),570-581.
[4] Shen Guang-Yu, Graded modules of graded Lie algebras of Cartan type, II, Scientica Sinica 29(1986), 1009-1019.
[5] Shen Guang-Yu, Graded modules of graded Lie algebras of Cartan type. III, Chinese Ann. Math. Ser.B 9(1988), 404-417 .
[6] Holmes R. R., Simple modules with character height at most one for the restricted Witt algebras, J.Algebra 237 No.2(2001), 446-469 .
[7] 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 .
[8] Wilson R. L., Automorphisms of graded Lie algebras of Cartan type, Comm.Algebra 3 No. 7(1975), 591-613.
[9] 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."
[10] Goodman- Wallach,Representations and Invariants of the Classical Groups, Cambridge University ,296-347.
[11] Peter Johannes Steffensen, Irreducible representations of the Witt-Jacobson Lie algebra of rank 2, Institute of Mathematics rAarhus University.
[12]R.E.Block and R.L.Wilson,Classification of the restricted simple Lie algebras,J.Algebra 112(1988),115-259.
[13]N.Jacobson,Classes of restricted Lie algebras of characteristic p,Ⅱ,Duke Math.J.10(1943),107-121.
[14]Jantzen,Jens Carsten,Representations of algebrauc groups,Library of Congress Cataloging-in-Publication Data,247-248.
[15]Hu Nai-Hong,The graded modules for the graded contact Cartan Lie algebras,Comm.Algebra 22(1994),4474-4497.
[16]Hu Nai-Hong,Irreducible constituents of graded modules for graded contact Lie algebras of Cartan type,Comm.Algebra 22(1994),5951-5970.
[2] Benkart G.M, Cartan subalgebras in Lie algebras of cartan type, Lie Algebras and Related Topics, CMS Conf. proc 5 (1986), 157-187.
[3] Shen Guang-Yu, Graded modules of graded Lie algebras of Cartan type, I ,Scientica Sinica 29(1986),570-581.
[4] Shen Guang-Yu, Graded modules of graded Lie algebras of Cartan type, II, Scientica Sinica 29(1986), 1009-1019.
[5] Shen Guang-Yu, Graded modules of graded Lie algebras of Cartan type. III, Chinese Ann. Math. Ser.B 9(1988), 404-417 .
[6] Holmes R. R., Simple modules with character height at most one for the restricted Witt algebras, J.Algebra 237 No.2(2001), 446-469 .
[7] 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 .
[8] Wilson R. L., Automorphisms of graded Lie algebras of Cartan type, Comm.Algebra 3 No. 7(1975), 591-613.
[9] 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."
[10] Goodman- Wallach,Representations and Invariants of the Classical Groups, Cambridge University ,296-347.
[11] Peter Johannes Steffensen, Irreducible representations of the Witt-Jacobson Lie algebra of rank 2, Institute of Mathematics rAarhus University.
[12]R.E.Block and R.L.Wilson,Classification of the restricted simple Lie algebras,J.Algebra 112(1988),115-259.
[13]N.Jacobson,Classes of restricted Lie algebras of characteristic p,Ⅱ,Duke Math.J.10(1943),107-121.
[14]Jantzen,Jens Carsten,Representations of algebrauc groups,Library of Congress Cataloging-in-Publication Data,247-248.
[15]Hu Nai-Hong,The graded modules for the graded contact Cartan Lie algebras,Comm.Algebra 22(1994),4474-4497.
[16]Hu Nai-Hong,Irreducible constituents of graded modules for graded contact Lie algebras of Cartan type,Comm.Algebra 22(1994),5951-5970.