Algebraic Approach to q,t-Characters

Details Algebraic Approach to q,t-Characters Algebraic Approach to q,t-Characters

2002-12-18

arxiv.org/abs/math/0212257v5

Advances in Mathematics 187 (2004), no. 1, 1--52

Mathematics

Quantum Algebra

46 pages; accepted for publication in Advances in Mathematics

Scientific paper

Frenkel and Reshetikhin introduced q-characters to study finite dimensional representations of quantum affine algebras. In the simply laced case Nakajima defined deformations of q-characters called q,t-characters. The definition is combinatorial but the proof of the existence uses the geometric theory of quiver varieties which holds only in the simply laced case. In this article we propose an algebraic general (non necessarily simply laced) new approach to q,t-characters motivated by our deformed screening operators. The t-deformations are naturally deduced from the structure of the quantum affine algebra: the parameter t is analog to the central charge c. The q,t-characters lead to the construction of a quantization of the Grothendieck ring and to general analogues of Kazhdan-Lusztig polynomials in the same spirit as Nakajima did for the simply laced case.

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