Algebraic approach to q, t-characters

Abstract

Frenkel and Reshetikhin (in: Recent Developments in Quantum Affine Algebras and Related Topics, Contemporary Mathematics, Vol. 248, 1999, pp. 163–205) introduced q-characters to study finite dimensional representations of the quantum affine algebra U q( g ̂ ) . In the simply laced case Nakajima (in: Physics and Combinatorics, Proceedings of the Nagoya 2000 International Workshop, World Scientific, Singapore, 2001, pp. 181–212; Preprint arXiv:math.QA/0105173) 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 the deformed screening operators (Internat. Math. Res. Not. 2003 (8) (2003) 451). The t-deformations are naturally deduced from the structure of U q( g ̂ ) : the parameter t is analog to the central charge c∈ U q( g ̂ ) . 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.