TY - JOUR
T1 - Q-systems as cluster algebras II
T2 - Cartan Matrix of Finite Type and the Polynomial Property
AU - Francesco, Philippe Di
AU - Kedem, Rinat
N1 - Funding Information:
We thank Bernhard Keller, Hugh Thomas and especially Sergei Fomin for their valuable input. RK thanks CEA-Saclay IPhT for their hospitality. We thank the organizers of the MSRI program on “Combinatorial Representation Theory” for their hospitality. RK is supported by NSF grant DMS-05-00759. PDF acknowledges the support of European Marie Curie Research Training Networks ENIGMA MRT-CT-2004-5652, ENRAGE MRTN-CT-2004-005616, ESF program MISGAM, and of ANR program GIMP ANR-05-BLAN-0029-01.
PY - 2009/10
Y1 - 2009/10
N2 - We define the cluster algebra associated with the Q-system for the Kirillov-Reshetikhin characters of the quantum affine algebra Uq(g{fraktur}̂) for any simple Lie algebra, generalizing the simply-laced case treated in (Kedem in Q-systems as cluster algebras. 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.
AB - We define the cluster algebra associated with the Q-system for the Kirillov-Reshetikhin characters of the quantum affine algebra Uq(g{fraktur}̂) for any simple Lie algebra, generalizing the simply-laced case treated in (Kedem in Q-systems as cluster algebras. 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.
KW - Cluster algebras
KW - Kirillov-Reshetikhin modules
KW - Q-systems
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U2 - 10.1007/s11005-009-0354-z
DO - 10.1007/s11005-009-0354-z
M3 - Article
AN - SCOPUS:70350392807
VL - 89
SP - 183
EP - 216
JO - Letters in Mathematical Physics
JF - Letters in Mathematical Physics
SN - 0377-9017
IS - 3
ER -