The q-Characters at Roots of Unity

Frenkel, Edward; Mukhin, E.

doi:10.1006/aima.2002.2084

Cited by 21 publications

(30 citation statements)

References 14 publications

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“…By construction, we have an isomorphism, (2) X r = A r (r ≥ 2). We remark that, in addition to T It is important to establish a precise relation between the ring Ch ℓ U q (ĝ) and the ring Rep U res ǫ (ĝ) of [FM2] for a primitive 2t(h ∨ + ℓ)th root of unity ǫ. A similar remark is applicable to the twisted quantum affine algebras as well.…”

Section: Remark On Periodicity Of Q-charactersmentioning

confidence: 99%

Periodicities of T-systems and Y-systems

Inoue¹,

Iyama²,

Kuniba³

et al. 2010

Nagoya Math. J.

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Abstract. The unrestricted T-system is a family of relations in the Grothendieck ring of the category of the finite-dimensional modules of the Yangian or the quantum affine algebra associated with a complex simple Lie algebra. The unrestricted T-system admits a reduction called the restricted T-system. In this paper we formulate the periodicity conjecture for the restricted T-systems, which is the counterpart of the known and partially proved periodicity conjecture for the restricted Y-systems. Then, we partially prove the conjecture by various methods: the cluster algebra and cluster category method for the simply laced case, the determinant method for types A and C, and the direct method for types A, D, and B (level 2).

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Section: Remark On Periodicity Of Q-charactersmentioning

confidence: 99%

Periodicities of T-systems and Y-systems

Inoue¹,

Iyama²,

Kuniba³

et al. 2010

Nagoya Math. J.

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“…There exists a basis of V (Y ,a Y −1 0,aq ) indexed by T [1, ] such that the action of U q (ŝl ∞ ) on it is given by formulas (7).…”

Section: Theorem 39mentioning

confidence: 99%

“…The quantum affinizations, in particular the quantum affine algebras and the quantum toroidal algebras, have been intensively studied (see for example [2,5,7,8,[10][11][12]21] and references therein). In recent works [19,20] we constructed new families of integrable representations of the quantum toroidal algebra U q (sl tor n+1 ), called extremal loop weight modules, which generalize the -highest weight modules: these are representations generated by an extremal vector for the horizontal quantum affine subalgebra in the sense of Kashiwara [13,15].…”

mentioning

confidence: 99%

Extremal Loop Weight Modules for U q ( sl ̂ ∞ ) $\mathcal {U}_{q}(\hat {sl}_{\infty })$

Mansuy

2015

Algebr Represent Theor

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We construct by fusion product new families of irreducible representations of the quantum affinization U q (ŝl ∞ ). The action is defined via the Drinfeld coproduct and is related to the crystal structure of semi-standard tableaux of type A ∞ . We call these representations extremal loop weight modules. The main motivations are applications to quantum toroidal algebras U q (sl tor n+1 ): we prove the conjectural link between U q (ŝl ∞ ) and U q (sl tor n+1 ) stated in Hernandez (J. Algebra 329, [147][148][149][150][151][152][153][154][155][156][157][158][159][160][161][162] 2011) for these families of representations. We recover in this way the extremal loop weight modules obtained in Mansuy (arXiv:1305.3481).

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“…Then, generalizing the argument of (1) It is important to establish a precise relation between the ring Ch U q (ĝ) and the ring Rep U res (ĝ) of [FM2] for a primitive 2t(h ∨ + )th root of unity . A similar remark is applicable to the twisted quantum affine algebras as well.…”

Section: Periodicities Of Restricted T-and Y-systems At Levelmentioning

confidence: 99%

Periodicities of T-systems and Y-systems

Inoue¹,

Iyama²,

Kuniba³

et al. 2010

Nagoya Mathematical Journal

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Abstract. The unrestricted T-system is a family of relations in the Grothendieck ring of the category of the finite-dimensional modules of Yangian or quantum affine algebra associated with a complex simple Lie algebra. The unrestricted T-system admits a reduction called the restricted T-system. In this paper we formulate the periodicity conjecture for the restricted T-systems, which is the counterpart of the known and partially proved periodicity conjecture for the restricted Y-systems. Then, we partially prove the conjecture by various methods: the cluster algebra and cluster category method for the simply laced case, the determinant method for types A and C, and the direct method for types A, D, and B (level 2). §1. IntroductionThe Y-system was introduced as a system of functional relations concerning the solutions of the thermodynamic Bethe ansatz equations for the factorizable scattering theory and the solvable lattice models [Z] [KNT]. As a side remark, it may be worth mentioning at this point that "T" stands for transfer matrix and "Q" stands for quantum character [Ki2] in the original literature.As a more recent development, a connection between the Q-systems and cluster algebras is clarified by [Ked], [DiK]. Also, a connection between the T-systems (or q-characters) and cluster algebras is studied while seeking a natural categorification of cluster algebras by abelian monoidal categories [HL].Having these results as a background, we make three simple observations:(1) There are actually two classes of the Y-systems (resp. T-systems), namely, the unrestricted and restricted Y-systems (resp. T-systems). The latter is obtained by a certain reduction from the former. The periodicity property above mentioned is for the restricted Y-systems. (2) The cluster algebra structure is simpler in the T-systems than in the Y-systems. (3) The representation theory of quantum affine algebras is more directly connected with the T-systems than with the Y-systems.These observations motivate us to ask if there is a similar periodicity property for the restricted T-systems, and indeed, there is.In this paper, we formulate the periodicity conjecture for the restricted T-systems, which is the counterpart of the known and partially proved periodicity conjecture for the restricted Y-systems. Then, we partially prove the conjecture by various methods. We remark that the restricted T-systems are relations in certain quotients of the Grothendieck ring Rep U q (ĝ), while the T-systems studied in [HL] are relations in certain subrings of Rep U q (ĝ). Accordingly, the correspondence between the T-systems for the simply laced case and cluster algebras considered here and the one in [HL] are close but slightly different. We also note that the correspondence between the unrestricted T-systems for the simply laced case and cluster algebras is described in [DiK, Appendix B].Let us explain the outline of the paper, whose contents are roughly divided into three parts.In the first part (Section 2) we introduce the unrestricted T-systems toget...

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The q-Characters at Roots of Unity

Abstract: We consider various specializations of the untwisted quantum affine algebras at roots of unity. We define and study the q-characters of their finite-dimensional representations. # 2002 Elsevier Science (USA)

Cited by 21 publications

References 14 publications

Periodicities of T-systems and Y-systems

Periodicities of T-systems and Y-systems

Extremal Loop Weight Modules for U q ( sl ̂ ∞ ) $\mathcal {U}_{q}(\hat {sl}_{\infty })$

Periodicities of T-systems and Y-systems

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