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\(t\)-analogue of the \(q\)-characters of finite-dimensional representations of quantum affine algebras. (English) Zbl 1011.17013

Kirillov, A. N. (ed.) et al., Physics and combinatorics. Proceedings of the Nagoya 2000 2nd international workshop, Nagoya, Japan, August 21-26, 2000. Singapore: World Scientific. 196-219 (2001).
Summary: E. Frenkel and N. Reshetikhin [Contemp. Math. 248, 163-205 (1999; Zbl 0973.17015)] introduced \(q\)-characters of finite-dimensional representations of quantum affine algebras. We give a combinatorial algorithm to compute them for all simple modules. Our tool is a \(t\)-analogue of the \(q\)-characters, which is similar to Kazhdan-Lusztig polynomials, and our algorithm has a resemblance with their definition. We need the theory of quiver varieties for the definition of \(t\)-analogues and the proof. But it appears only in the last section. The rest of the paper is devoted to an explanation of the algorithm, which one can read without the knowledge about quiver varieties. A proof is given only in part. A full proof is to appear elsewhere.
For the entire collection see [Zbl 0964.00051].

MSC:

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

Citations:

Zbl 0973.17015
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