ar X iv : 0 90 1 . 32 05 v 2 [ m at h . R T ] 1 9 Ju n 20 09 DOUBLE AFFINE LIE ALGEBRAS AND FINITE GROUPS
نویسنده
چکیده
We begin to study the Lie theoretical analogs of symplectic reflection algebras for Γ a finite cyclic group, which we call “cyclic double affine Lie algebra”. We focus on type A : in the finite (resp. affine, double affine) case, we prove that these structures are finite (resp. affine, toroidal) type Lie algebras, but the gradings differ. The case which is essentially new is sln(C[u, v]⋊Γ). We describe its universal central extensions and start the study of its representation theory, in particular of its highest weight integrable modules and Weyl modules. We also consider the first Weyl algebra A1 instead of the polynomial ring C[u, v], and, more generally, a rank one rational Cherednik algebra. We study quasi-finite highest weight representations of these Lie algebras.
منابع مشابه
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Multiloop algebras determined by n commuting algebra automor-phisms of finite order are natural generalizations of the classical loop algebras that are used to realize affine Kac-Moody Lie algebras. In this paper, we obtain necessary and sufficient conditions for a Z n-graded algebra to be realized as a multiloop algebra based on a finite dimensional simple algebra over an algebraically closed ...
متن کاملar X iv : 0 90 4 . 04 99 v 3 [ m at h . R T ] 5 M ay 2 00 9 DEGENERATE AFFINE HECKE - CLIFFORD ALGEBRAS AND TYPE Q LIE SUPERALGEBRAS
We construct the finite dimensional simple integral modules for the (degenerate) affine Hecke-Clifford algebra (AHCA), H aff Cℓ (d). Our construction includes an analogue of Zelevin-sky's segment representations, a complete combinatorial description of the simple calibrated H aff Cℓ (d)-modules, and a classification of the simple integral H aff Cℓ (d)-modules. Our main tool is an analogue of th...
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