Representations of Double Affine Lie algebras (original) (raw)

Double affine Lie algebras and finite groups

Pacific Journal of Mathematics, 2009

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 A 1 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.

Quantum affine algebras and their representations

Journal of Pure and Applied Algebra, 1995

We prove a highest weight classification of the finite-dimensional irreducible representations of a quantum affine algebra, in the spirit of Cartan's classification of the finite-dimensional irreducible representations of complex simple Lie algebras in terms of dominant integral weights. We also survey what is currently known about the structure of these representations.

Integrable and Weyl Modules for Quantum Affine sl2

Quantum Groups and Lie Theory, 2002

Let t be an arbitrary symmetrizable Kac-Moody Lie algebra and U q (t) the corresponding quantized enveloping algebra of t defined over C(q). If µ is a dominant integral weight of t then one can associate to it in a natural way an irreducible integrable U q (t)-module L(µ). These modules have many nice properties and are well understood,

Integrable representations of the quantum affine special linear superalgebra

The simple integrable modules with finite dimensional weight spaces are classified for the quantum affine special linear superalgebra Uq(hatmathfraksl(M∣N))\U_q(\hat{\mathfrak{sl}}(M|N))Uq(hatmathfraksl(M∣N)) at generic qqq. Any such module is shown to be a highest weight or lowest weight module with respect to one of the two natural triangular decompositions of the quantum affine superalgebra depending on whether the level of the module is zero or not. Furthermore, integrable Uq(hatmathfraksl(M∣N))\U_q(\hat{\mathfrak{sl}}(M|N))Uq(hatmathfraksl(M∣N))-modules at nonzero levels exist only if MMM or NNN is 111.

Representations of the generalized Lie algebra

Journal of Physics A: Mathematical and General, 1998

We construct finite-dimensional irreducible representations of two quantum algebras related to the generalized Lie algebra sl(2) q introduced by Lyubashenko and the second named author. We consider separately the cases of q generic and q at roots of unity. Some of the representations have no classical analog even for generic q. Some of the representations have no analog to the finite-dimensional representations of the quantised enveloping algebra U q (sl(2)), while in those that do there are different matrix elements.