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\(Q\)-systems as cluster algebras. II: Cartan matrix of finite type and the polynomial property. (English) Zbl 1195.81077

Summary: We define the cluster algebra associated with the \(Q\)-system for the Kirillov-Reshetikhin characters of the quantum affine algebra \({U_q(\widehat{\mathfrak g})}\) for any simple Lie algebra \({\mathfrak g}\), generalizing the simply-laced case treated in part I [R. Kedem, J. Phys. A, Math. Theor. 41, No. 11, Article ID194011, 14 p. (2008; Zbl 1141.81014); arXiv:0712.2695[math.RT] (2007)]. We describe some special properties of this cluster algebra, and explain its relation to the deformed \(Q\)-systems which appeared on our proof of the combinatorial-KR conjecture. We prove that the polynomiality of the cluster variables in terms of the “initial cluster seeds”, including solutions of the \(Q\)-system, is a consequence of the Laurent phenomenon and the boundary conditions. We also define the cluster algebra associated with \(T\)-systems, or general systems which take the form of \(T\)-systems in the bipartite case. Such systems describe the recursion relations satisfied by the \(q\)-characters of Kirillov-Reshetikhin modules and also appear in the categorification picture in terms of preprojective algebras of Geiss, Leclerc and Schröer. We give a formulation of both \(Q\)-systems and generalized \(T\)-systems as cluster algebras with coefficients. This provides a proof of the polynomiality of solutions of all such “generalized \(T\)-systems” with appropriate boundary conditions.

MSC:

81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
05E15 Combinatorial aspects of groups and algebras (MSC2010)
17B37 Quantum groups (quantized enveloping algebras) and related deformations
13F60 Cluster algebras
20C20 Modular representations and characters

Citations:

Zbl 1141.81014
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References:

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