Kac-Moody代数在U(h)上的模结构
|
摘要
令g是任意Kac-Moody代数,其Cartan子代数为h.本文确定了在U(h)上限制是秩1自由的g-模组成的模范畴.确切地讲,这个模范畴是非空的当且仅当g是A_l型或是C_l型的,其中l是正整数.
Let g be an arbitrary Kac-Moody algebra with a Cartan subalgebra h. In this paper, we determine the category of g-modules that are free U(h)-modules of rank 1. More precisely, this category of g-modules is not empty if and only if g is of type A_l or C_l for any positive integer l.
Let g be an arbitrary Kac-Moody algebra with a Cartan subalgebra h. In this paper, we determine the category of g-modules that are free U(h)-modules of rank 1. More precisely, this category of g-modules is not empty if and only if g is of type A_l or C_l for any positive integer l.
引文
1 Block R.The irreducible representations of the Lie algebra sl(2)and of the Weyl algebra.Adv Math,1981.139:69-110
2 Mathieu O.Classification of irreducible weight modules.Ann Inst Fourier(Grenoble),2000,50:537-592
3 Dimitrov I,Grantcharov D.Classification of simple weight modules over affine Lie algebras.ArXiv:0910.0688,2009
4 Futorny V,Tsylke A.Classification of irreducible nonzero level modules with finite-dimensional weight spaces for affine Lie algebras.J Algebra,2001,238:426-441
5 Adamovic D,L(u|¨)R,Zhao K.Whittaker modules for the affine Lie algebra A(1)1.Adv Math,2016,289:438-479
6 Bekkert V,Benkart G,Futorny V,et al.New irreducible modules for Heisenberg and affine Lie algebras.J Algebra,2013,373:284-298
7 Christodoulopoulou K.Whittaker modules for Heisenberg algebras and imaginary Whittaker modules for affine Lie algebras.J Algebra,2008,320:2871-2890
8 Futorny V,Grantcharov D,Martins R.Localization of free field realizations of affine Lie algebras.Lett Math Phys,2015,105:483-502
9 Guo X,Zhao K.Irreducible representations of non-twisted affine Kac-Moody algebras.ArXiv:1305.4059,2013
10 Liu G,Zhao Y.IrreducibleA(1)1-modules from modules over two-dimensional non-abelian Lie algebra.Front Math China,2016,11:353-363
11 Mazorchuk V,Zhao K.Characterization of simple highest weight modules.Canad Math Bull,2013,56:606-614
12 Rao S.Classification of loop modules with finite-dimensional weight spaces.Math Ann,1996,305:651-663
13 Rao S.Classification of irreducible integrable modules for multi-loop algebras with finite-dimensional weight spaces.J Algebra,2001,246:215-225
14 Nilsson J.Simple sl_(n+1)-module structures on U(h).J Algebra,2015,424:294-329
15 Cai Y,Liu G,Nilsson J,et al.Generalized Verma modules over sl_(n+2) induced from U(h_n)-free sl_(n+1)-modules.ArXiv:1510.04129,2015
16 Nilsson J.U(h)-free modules and coherent families.J Pure Appl Algebra,2016,220:1475-1488
17 Tan H,Zhao K.W_n~+-and W_n-module structures on U(h_n).J Algebra,2015,424:357-375
18 Chen H,Guo X.Modules over the Heisenberg-Virasoro and W(2,2).ArXiv:1401.4670,2014
19 Cai Y,Zhao K.Module structure on U(H)for basic Lie superalgebras.Toyama Math J,2015,37:55-72
20 Kac V.Infinite Dimensional Lie Algebras,3rd ed.Cambridge:Cambridge University Press,1990
21 Humphreys J.Introduction to Lie Algebras and Representation Theory.Graduate Texts in Mathematics,vol.9.New York-Berlin:Springer-Verlag,1972
2 Mathieu O.Classification of irreducible weight modules.Ann Inst Fourier(Grenoble),2000,50:537-592
3 Dimitrov I,Grantcharov D.Classification of simple weight modules over affine Lie algebras.ArXiv:0910.0688,2009
4 Futorny V,Tsylke A.Classification of irreducible nonzero level modules with finite-dimensional weight spaces for affine Lie algebras.J Algebra,2001,238:426-441
5 Adamovic D,L(u|¨)R,Zhao K.Whittaker modules for the affine Lie algebra A(1)1.Adv Math,2016,289:438-479
6 Bekkert V,Benkart G,Futorny V,et al.New irreducible modules for Heisenberg and affine Lie algebras.J Algebra,2013,373:284-298
7 Christodoulopoulou K.Whittaker modules for Heisenberg algebras and imaginary Whittaker modules for affine Lie algebras.J Algebra,2008,320:2871-2890
8 Futorny V,Grantcharov D,Martins R.Localization of free field realizations of affine Lie algebras.Lett Math Phys,2015,105:483-502
9 Guo X,Zhao K.Irreducible representations of non-twisted affine Kac-Moody algebras.ArXiv:1305.4059,2013
10 Liu G,Zhao Y.IrreducibleA(1)1-modules from modules over two-dimensional non-abelian Lie algebra.Front Math China,2016,11:353-363
11 Mazorchuk V,Zhao K.Characterization of simple highest weight modules.Canad Math Bull,2013,56:606-614
12 Rao S.Classification of loop modules with finite-dimensional weight spaces.Math Ann,1996,305:651-663
13 Rao S.Classification of irreducible integrable modules for multi-loop algebras with finite-dimensional weight spaces.J Algebra,2001,246:215-225
14 Nilsson J.Simple sl_(n+1)-module structures on U(h).J Algebra,2015,424:294-329
15 Cai Y,Liu G,Nilsson J,et al.Generalized Verma modules over sl_(n+2) induced from U(h_n)-free sl_(n+1)-modules.ArXiv:1510.04129,2015
16 Nilsson J.U(h)-free modules and coherent families.J Pure Appl Algebra,2016,220:1475-1488
17 Tan H,Zhao K.W_n~+-and W_n-module structures on U(h_n).J Algebra,2015,424:357-375
18 Chen H,Guo X.Modules over the Heisenberg-Virasoro and W(2,2).ArXiv:1401.4670,2014
19 Cai Y,Zhao K.Module structure on U(H)for basic Lie superalgebras.Toyama Math J,2015,37:55-72
20 Kac V.Infinite Dimensional Lie Algebras,3rd ed.Cambridge:Cambridge University Press,1990
21 Humphreys J.Introduction to Lie Algebras and Representation Theory.Graduate Texts in Mathematics,vol.9.New York-Berlin:Springer-Verlag,1972