Quantum affine algebras

Chari, Vyjayanthi; Pressley, Andrew

doi:10.1007/bf02102063

Cited by 316 publications

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“…For example, for X r = A r , the relations in (2.2) are the Jacobi identities among the Jacobi-Trudi-type determinantal expression of the transfer matrices in [BR]. The T-system is a natural affinization of the Q-system of [Ki, KR] (see Appendix A.1), and the idea behind the both systems was the existence of a conjectured family of exact sequences among the Kirillov-Reshetikhin modules [KR,CP1,KNS1] of the Yangian Y (g) and/or the untwisted quantum affine algebra U q (ĝ) associated with the complex simple Lie algebra g of type X r [D1, D2, J]. The choice U = C corresponds to the Y (g) case, while the choice U = C ξ corresponds to the U q (ĝ) case as explained below.…”

Section: Unrestricted T and Y-systemsmentioning

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: Unrestricted T and Y-systemsmentioning

confidence: 99%

Periodicities of T-systems and Y-systems

Inoue¹,

Iyama²,

Kuniba³

et al. 2010

Nagoya Math. J.

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“…The lemma will follow by calculating the images of the imaginary root vectors under F r. It follows from [CP1] that modulo the ideal in U 1 generated by the real root vectors, E kδ is primitive for each k ∈ Z. For the same reason (and since the center is closed under coproduct by the previous Corollary) it follows that E k δ is primitive modulo the -th powers of the real root vectors.…”

Section: Corollary 321 F R Is a Hopf Algebra Homomorphismmentioning

confidence: 93%

Finite-dimensional representations of quantum affine algebras at roots of unity

Beck

Kač

1996

J. Amer. Math. Soc.

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The purpose of this paper is to study finite-dimensional irreducible representations of the quantum loop algebra U ε = U ε ( g) at an odd root of unity ε. Here g is a simple finite-dimensional Lie algebra over C and g = C[t, t −1 ] ⊗ C g is the associated loop algebra.Denoting by Spec R the set of all finite-dimensional irreducible complex representations of an associative algebra R over C and by Z the center of R, we have (by Schur's lemma) the canonical map:(Recall that the value of χ on a representation σ ∈ Spec R is defined by σ(z) = χ(σ)I for z ∈ Z.) If R is a finitely generated module over Z (which is the case for R = U ε (g) [DC-K]) one knows that the map χ is surjective with finite fibers and, moreover, it is bijective over a Zariski open dense subset of Spec Z. In other words, at least "generically", Spec Z parametrizes the set of all irreducible finitedimensional irreducible representations of R. This well-known observation was the starting point for a thorough (albeit incomplete) study of Spec U ε (g) taken up in [DC-K], [DC-K-P1,2,3] and other papers. In the case when R = U ε the situation is quite different since U ε is not finitely generated over its center Z = Z ε . The canonical map χ is not surjective and is not generically bijective. The main result of the present paper is the calculation of the image of χ in Spec Z ε .The first result (Proposition 2.3) provides a (infinite) set of generators of the algebra Z ε . By general principles, Z ε has a canonical structure of a Poisson algebra. Furthermore, we show that Z ε is a Hopf subalgebra of the Hopf algebra U ε . (Recall that this isn't the case for U ε (g).) Thus, Z ε is a Poisson Hopf algebra, and using a "Frobenius homomorphism" we obtain that it is isomorphic to a certain Poisson Hopf algebra U 1 independent of the odd root of unity ε (Corollaries 3.2.1 and 3.2.2).In the dual language, Spec Z ε is a Poisson proalgebraic group. Our first key result (Theorem 5.3) is the construction of a Poisson group isomorphism (0.2) π : Spec Z ε → Ω

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“…Proof We follow the proof of [ [13], Proposition 3.2] step by step. From the finite-dimensional representation of the commutative algebra A M,N on V , one finds λ ∈ P M,N such that…”

Section: Lemma 412 Suppose M N > 0 and (M Nmentioning

confidence: 99%

Representations of quantum affine superalgebras

Zhang

2014

Math. Z.

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We study the quantum affine superalgebra U q (Lsl(M, N )) and its finitedimensional representations. We prove a triangular decomposition and establish a system of Poincaré-Birkhoff-Witt generators for this superalgebra, both in terms of Drinfel'd currents. We define the Weyl modules in the spirit of Chari-Pressley and prove that these Weyl modules are always finite-dimensional and non-zero. In consequence, we obtain a highest weight classification of finite-dimensional simple representations when M = N . Some concrete simple representations are constructed via evaluation morphisms.

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Quantum affine algebras

Cited by 316 publications

References 6 publications

Periodicities of T-systems and Y-systems

Periodicities of T-systems and Y-systems

Finite-dimensional representations of quantum affine algebras at roots of unity

Representations of quantum affine superalgebras

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