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

UR - http://www.scopus.com/inward/record.url?scp=70350392807&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=70350392807&partnerID=8YFLogxK

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 -