### Citations

1050 | Pressley.A : ” A guide to Quantum Groups - Chari - 1994 |

238 |
A new realization of Yangians and quantized affine algebras
- Drinfeld
- 1988
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Citation Context ...nomial: [n]q := qn − q−n q − q−1 , [n]q! := [n]q [n− 1]q . . . [1]q , [ n m ] q := [n]q! [n−m]q! [m]q! . 2.2. Quantum Affine Algebras. The quantum affine algebra Uq(ĝ) in Drinfeld’s new realization, =-=[Dri88]-=- is generated by x±i,n (i ∈ I, n ∈ Z), k ±1 i (i ∈ I), hi,n (i ∈ I, n ∈ Z\{0}) and central elements c±1/2, subject to the following relations: kikj = kjki, kihj,n = hj,nki, kix ± j,nk −1 i = q ±Bijx±j... |

136 |
Quantum affine algebras
- Chari, Pressley
- 1991
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Citation Context ...aq−1(m) ≤ u1,aq(m), mA −1 1,a ∈ M (W (M)) \M (L(M)); (iii) 0 ≥ u1,aq−1(m), mA −1 1,a /∈ M (W (M)). Proof. The result follows from the known closed forms of all Weyl modules [CP01] and simple modules, =-=[CP91]-=-, in type A1. Proof of Theorem 2.1. For convenience, let us define χn := truncm+Q−U,(=n) χq(L(m+)) Mn := truncm+Q−U,(=n) (M), (B.1) and similarly χ≤n and M≤n. Given property (i), to prove the equali... |

122 | Functional relations in solvable lattice models. 1: Functional relations and representation theory - Kuniba, Nakanishi, et al. - 1994 |

110 | The q-characters of of representation of quantum affine algebras and deformations of W-algebras
- Frenkel, N
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Citation Context ...decomposition (2.3) of a finite-dimensional representation V into its Uq(g)-weight spaces can be refined by decomposing it into the Jordan subspaces of the mutually commuting φ±i,±r defined in (2.2), =-=[FR98]-=-: V = ⊕ γ Vγ , γ = (γ ± i,±r)i∈I,r∈Z≥0 , γ ± i,±r ∈ C (2.4) where Vγ = {v ∈ V : ∃k ∈ N, ∀i ∈ I,m ≥ 0 ( φ±i,±m − γ ± i,±m )k v = 0} . If dim(Vγ) > 0, γ is called an l-weight of V . For every finite-dim... |

89 | Quantum affine algebras and their representations - Chari, A - 1994 |

87 | Weyl modules for classical and quantum affine algebras
- Chari, A
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Citation Context ... there exists a highest l-weight representation W (m), called the Weyl module, with the property that every highest l-weight representation of Uq(ĝ) with highest l-weight γ(m) is a quotient of W (m) =-=[CP01]-=-. A finite-dimensional Uq(ĝ)-module V is said to be special if and only if χq(V ) has exactly one dominant monomial. It is anti-special if and only if χq(V ) has exactly one anti-dominant monomial. I... |

75 | Combinatorics of q-characters of finite-dimensional representations of quantum affine algebras
- Frenkel, E
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Citation Context ...∗. Note that wtAi,a = αi. There is a partial order ≤ on P in which m ≤ m ′ if and only if m′m−1 ∈ Q+. It is compatible with the partial order on P in the sense that m ≤ m′ implies wtm ≤ wtm′. We have =-=[FM01]-=- that for all m+ ∈ P +, M (L(m+)) ⊂ m+Q −. (2.7) EXTENDED T-SYSTEMS 7 For all i ∈ I, a ∈ C∗ let ui,a be the homomorphism of abelian groups P → Z such that ui,a(Yj,b) = 1 i = j and a = b0 otherwise.... |

72 | Reshetikhin, “Representations of Yangians and multiplicities of the inclusion of the irreducible components of the tensor product of representations of simple Lie algebras - Kirillov, Y - 1990 |

71 | Cluster algebras and quantum affine algebras
- Hernandez, Leclerc
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Citation Context ...eralizing the results of [Cha95]. In types ADE, there is a recent remarkable conjecture on the cluster algebra relations in the category of finite-dimensional representations of affine quantum groups =-=[HL10]-=- which includes the T-system. We should like to think that similar to the T-systems, the extended T-systems would provide a large family of explicit cluster relations in the cluster algebra. The paper... |

64 | Representations of Yangians with Gelfand-Zetlin bases
- Nazarov
- 1998
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Citation Context ...resentations. We proceed to extend our T-system to yet a larger class of modules – the snake modules, introduced in [MY]. In type A, the snake modules are just modules related to skew Young diagrams, =-=[NT98]-=-. In type B, the modules related to skew Young diagrams [KOS95] form a subset of snake modules. The term “snake” is meant to be suggestive of the pattern formed by the zeros of the Drinfeld polynomial... |

62 | t-analogs of q-characters of Kirillov-Reshetikhin modules of quantum affine algebras, Represent. Theory 7 - Nakajima - 2003 |

50 |
The Kirillov-Reshetikhin conjecture and solutions of T -systems
- Hernandez
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Citation Context ...Yangians and quantum affine algebras [KR90, KNS94, Nak03, Her06]. More precisely, the T-systems correspond to a family of short exact sequences of tensor products of Kirillov-Reshetikhin (KR) modules =-=[Her06]-=-. The knowledge of the T-system is one of the main reasons the KR modules are comparatively well understood. In this paper we argue that other classes of finite-dimensional modules of quantum affine a... |

45 | Minimal affinizations of representations of quantum groups: the rank 2 case, preprint
- Chari
- 1994
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Citation Context ...tems we consider the corresponding extended Qsystem of typeB2. We use it to compute the decomposition of B2 wrapping modules after restriction to the finite quantum group, generalizing the results of =-=[Cha95]-=-. In types ADE, there is a recent remarkable conjecture on the cluster algebra relations in the category of finite-dimensional representations of affine quantum groups [HL10] which includes the T-syst... |

33 | Beyond Kirillov-Reshetikhin modules, Quantum affine algebras, extended affine Lie algebras, and their applications
- Chari, Hernandez
- 2010
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Citation Context ...t=1 Yit,kt ) 6∼= L ∏ (i,k)∈U Yi,k ⊗ L ∏ (i,k)∈U ′ Yi,k . (6.4) The points of U may not form a snake but χq ( L (∏ (i,k)∈U Yi,k )) always includes the monomial∏ (i,k)∈U m(p − i,k): see e.g. =-=[CH10]-=- Theorem 3. Therefore, the q-character of the right-hand side contains the monomial m := ∏ (i,k)∈U m(p−i,k) ∏ (i,k)∈U ′ m(p+i,k). (6.5) We claim that the q-character of the left-hand side of (6.4) doe... |

27 |
On minimal affinizations of representations of quantum groups
- Hernandez
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Citation Context ...tify a suitable class of representations – which one would like to include all minimal affinizations – and to furnish the necessary proofs. Note that minimal affinizations in other types are not thin =-=[Her07]-=- in general, which makes the analysis more difficult. The explicit form of several instances of the extended T-system is written in Section 4. For example, the extended T-system for evaluation modules... |

26 | The Hopf algebra Rep Uq(ĝl - Frenkel, Mukhin |

26 | T-systems and Y-systems in integrable systems
- Kuniba, Nakanishi, et al.
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Citation Context .... 1. Introduction The T-systems are important sets of recurrence relations which have many applications in integrable systems. The literature on the subject is vast: we refer the reader to the survey =-=[KNS11]-=- and references therein. Originally, the T-systems were introduced as a family of relations in the Grothendieck ring of the category of the finite-dimensional modules of the Yangians and quantum affin... |

26 |
Quantum Jacobi–Trudi and Giambelli formulae for Uq(B(1)r ) from the analytic Bethe ansatz
- A, Ohta, et al.
- 1995
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Citation Context ...r class of modules – the snake modules, introduced in [MY]. In type A, the snake modules are just modules related to skew Young diagrams, [NT98]. In type B, the modules related to skew Young diagrams =-=[KOS95]-=- form a subset of snake modules. The term “snake” is meant to be suggestive of the pattern formed by the zeros of the Drinfeld polynomials of such modules, c.f. Figure 2. In this paper we work in type... |

22 |
New features of FORM,” arXiv:math-ph/0010025
- Vermaseren
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Citation Context ...10) and by the EPSRC (from Nov 2010, grant number EP/H000054/1). The research of EM is supported by the NSF, grant number DMS-0900984. Computer programs to calculate q-characters were written in FORM =-=[Ver]-=-. 2. Background 2.1. Cartan data. Let g be a complex simple Lie algebra of rank N and h a Cartan subalgebra of g. We identify h and h∗ by means of the invariant inner product 〈·, ·〉 on g normalized su... |

8 |
Factorization of representations of quantum affine algebras, Modular interfaces
- CHARI, PRESSLEY
- 1997
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Citation Context ...taneously diagonalizable with joint simple spectrum. A finite-dimensional Uq(ĝ)-module V is said to be prime if and only if it is not isomorphic to a tensor product of two non-trivial Uq(ĝ)-modules =-=[CP97]-=-. Let χ : Rep(Uq(g))→ Z[e ±ωi ]i∈I be the Uq(g)-character homomorphism. Let wt : P → P be the homomorphism of abelian groups defined by wt : Yi,a 7→ ωi. The map wt induces in an obvious way a map ZP →... |

5 |
Path description of type
- Mukhin, Young
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Citation Context ...which allows one to compute, in particular, the wrapping modules via fundamental representations. We proceed to extend our T-system to yet a larger class of modules – the snake modules, introduced in =-=[MY]-=-. In type A, the snake modules are just modules related to skew Young diagrams, [NT98]. In type B, the modules related to skew Young diagrams [KOS95] form a subset of snake modules. The term “snake” i... |

1 |
q-characters and minimal affinizations, Int
- Moakes, Pressley
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Citation Context ...imension does not factor fully to linear factors with integer coefficients. Some 3-term relations among minimal affinizations in type B2, different from the ones in the present paper, can be found in =-=[MP07]-=-. 22 E. MUKHIN AND C. A. S. YOUNG 6. Paths and moves In order to prove Theorem 4.1 we first recall from [MY] the closed form for the q-characters of snake modules in terms of non-overlapping paths. 6.... |