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Algebraic approach to \(q,t\)-characters. (English) Zbl 1098.17009
Summary: E. Frenkel and N. Reshetikhin [Contemp. Math. 248, 163–205 (1999; Zbl 0973.17015)] introduced \(q\)-characters to study finite dimensional representations of the quantum affine algebra \(\mathcal U_q(\hat{\mathfrak g})\). In the simply laced case H. Nakajima [Proceedings of the Nagoya 2000 2nd international workshop, Nagoya, Japan, 2000, 196–219 (2001; Zbl 1011.17013); see also 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 [Int. Math. Res. Not. 2003, No. 8, 451–475 (2003; Zbl 1098.17005)]. The \(t\)-deformations are naturally deduced from the structure of : the parameter \(t\) is analog to the central charge \(c \in \mathcal U_q(\hat{\mathfrak g})\). 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.

17B37 Quantum groups (quantized enveloping algebras) and related deformations
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
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