Abstract
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 (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.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Kirillov, A.N., Reshetikhin, N.Yu.: Formulas for the multiplicities of the occurrence of irreducible components in the tensor product of representations of simple Lie algebras, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 205 (1993), no. Differentsialnaya Geom. Gruppy Li i Mekh. 13, 30–37, 179
Hatayama G., Kuniba A., Okado M., Takagi T., Yamada Y.: Remarks on fermionic formula, Recent developments in quantum affine algebras and related topics (Raleigh, NC, 1998), Contemp. Math., vol. 248, pp. 243–291. Amer. Math. Soc., Providence, RI (1999)
Hatayama G., Kuniba A., Okado M., Takagi T., Tsuboi Z.: Paths, crystals and fermionic formulae, MathPhys Odyssey, 2001, Prog. Math. Phys., vol. 23, pp. 205–272. Birkhäuser Boston, Boston, MA (2002)
Hernandez, D.: Kirillov–reshetikhin conjecture: the general case. arXiv:0704.2838 [math.QA]
Atsuo K., Tomoki N.: Bethe equation at q = 0, Möbius inversion formula, and weight multiplicities: II. X n case. J. Algebra 251, 577–618 (2002)
Atsuo K., Tomoki N., Junji S.: Functional relations in solvable lattice models. I. Functional relations and representation theory. Internat J. Modern Phys. A 9(30), 5215–5266 (1994)
Hiraku N.: t-analogs of q-characters of Kirillov–Reshetikhin modules of quantum affine algebras. Represent. Theory 7, 259–274 (2003) (electronic)
Hernandez D.: The Kirillov–Reshetikhin conjecture and solutions of T-systems. J. Reine Angew. Math. 596, 63–87 (2006)
Sergey F., Andrei Z.: Cluster algebras. I. Foundations. J. Amer. Math. Soc. 15(2), 497–529 (2002)
Sergey F., Andrei Z.: Y-systems and generalized associahedra. Ann. of Math. (2) 158(3), 977–1018 (2003)
Kedem, R.: Q-systems as cluster algebras. arXiv:0712.2695 [math.RT] (2007)
Di Francesco, P., Kedem, R.: Proof of the combinatorial Kirillov–Reshetikhin conjecture arXiv:0710.4415 [math:QA] (IMRN, to appear)
Sergey F., Andrei Z.: The Laurent phenomenon. Adv. in Appl. Math. 28(2), 119–144 (2002)
Chari V., Pressley A.: A guide to quantum groups. Cambridge University Press, Cambridge (1995)
Sergey F., Andrei Z.: Cluster algebras. IV. Coefficients. Compos. Math. 143(1), 112–164 (2007)
Berenstein A., Fomin S., Zelevinsky A.: Cluster algebras III: upper bounds and double Bruhat cells. Duke Math. J. 126(1), 1–52 (2005)
Frenkel, E., Reshetikhin, N.: The q-characters of representations of quantum affine algebras and deformations of W-algebras. In: Recent developments in quantum affine algebras and related topics (Raleigh, NC 1998). Contemp. Math. 248, 163–205 (1999)
Geiss, C., Leclerc, B., Schröer, J.: Rigid modules over preprojective algebras ii: the Kac-Moody case, arXiv:math/0703039 [math.RT]
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Di Francesco, P., Kedem, R. Q-systems as Cluster Algebras II: Cartan Matrix of Finite Type and the Polynomial Property. Lett Math Phys 89, 183–216 (2009). https://doi.org/10.1007/s11005-009-0354-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11005-009-0354-z