Deformed Kac-Moody algebras and their representations

JOURNAL OF ALGEBRA

卷:319

Article

Liu, Jianbo²  Zhao, Kaiming¹ 

[1] Wilfrid Laurier Univ, Dept Math, Waterloo, ON N2L 3C5, Canada

[2] Chinese Acad Sci, Inst Math, Acad Math & Syst Sci, Beijing 100080, Peoples R China

关键词: deformed Kac-Moody algebra;    invariant symmetric bilinear form;    lowest (respectively highest) weight Verma module;    irreducible module;   

DOI  :  10.1016/j.jalgebra.2008.01.009

来源: Elsevier

【 摘 要 】

A class of Lie algebras S (A) associated to generalized Cartan matrices A is studied. The Lie algebras S(A) have much simpler structure than Kac-Moody algebras, but have the same root spaces with g(A). In particular, S(A) has an abelian subalgebra of half size. We show that, S(A) has a non-degenerate invariant symmetric bilinear form if and only if A is symmetrizable; S(X-1) congruent to S(X-2) if and only if the GCMs X-1 and X-2 are the same up to a permutation of rows and columns. We study the lowest (respectively highest) weight Verma module (V) over bar(lambda) (respectively (V) over tilde (lambda)) over S(A), and obtain the necessary and sufficient conditions for (V) over bar(lambda) to be irreducible, and also find its maximal proper submodule when (V) over bar(lambda) is reducible. Then using graded dual module of (V) over bar(lambda) we deduce the necessary and sufficient conditions for (V) over tilde (lambda) to be irreducible. (C) 2008 Elsevier Inc. All rights reserved.

【 授权许可】

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