Abstract
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.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 183-216 |
| Number of pages | 34 |
| Journal | Letters in Mathematical Physics |
| Volume | 89 |
| Issue number | 3 |
| DOIs | |
| State | Published - Oct 2009 |
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Keywords
- Cluster algebras
- Kirillov-Reshetikhin modules
- Q-systems
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics
Cite this
Q-systems as cluster algebras II : Cartan Matrix of Finite Type and the Polynomial Property. / Francesco, Philippe Di; Kedem, Rinat.
In: Letters in Mathematical Physics, Vol. 89, No. 3, 10.2009, p. 183-216.Research output: Contribution to journal › Article
}
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
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 -