### Citations

1050 |
Pressley.A : ” A guide to Quantum Groups
- Chari
- 1994
(Show Context)
Citation Context ..., t ∈ Z≥0, then v is called a highest l-weight vector with highest l-weight γ(m). The module V is called a highest l-weight representation if V = Uq ĝ · v for some highest l-weight vector v ∈ V . In =-=[CP94]-=-, [CP95a], it is shown that there is a bijection between the set of isomorphism classes of finite-dimensional irreducible highest l-weight Uqĝ-modules of type 1. Let L(m+) denote the irreducible high... |

136 |
Quantum affine algebras
- Chari, Pressley
- 1991
(Show Context)
Citation Context ...ny nonhighest monomial in M (L(m+)) is right-negative and hence L(m+) is special. 2.4. q-characters of Uqŝl2-modules and the FM algorithm. The q-characters of Uqŝl2-modules are well-understood, see =-=[CP91]-=-, [FR98]. We recall the results here. Let W (a) k be the irreducible representation Uqŝl2 with highest weight monomial X (a) k = k−1∏ i=0 Yaqk−2i−1 , where Ya = Y1,a. Then the q-character of W (a) k ... |

122 | Functional relations in solvable lattice models. 1: Functional relations and representation theory
- Kuniba, Nakanishi, et al.
- 1994
(Show Context)
Citation Context ...tion The T-systems are some families of relations in the Grothendieck ring of the category of the finite-dimensional modules of quantum affine algebras (or Yangians), see [K83], [K84], [K87], [KR90], =-=[KNS94]-=-, [Nak03], [Her06]. The T-systems are widely applied to representation theory, combinatorics and integrable systems, see the recent survey [KNS11]. The modules in the usual T-systems are the Kirillov-... |

113 |
A new realization of Yangians and quantum affine algebras
- Drinfeld
- 1988
(Show Context)
Citation Context ...N-RONG LI The quantum affine algebra Uqĝ is a C(q)-algebra 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 certain relations, see =-=[Dri88]-=-. The algebra Uqĝ is a Hopf algebra. Let Uqg be the quantized enveloping algebra of g. The subalgebra of Uqĝ generated by (k±i )i∈I , (x ± i,0)i∈I is a Hopf subalgebra of Uqĝ and is isomorphic as a... |

89 | Quantum affine algebras and their representations
- Chari, A
- 1994
(Show Context)
Citation Context ...0, then v is called a highest l-weight vector with highest l-weight γ(m). The module V is called a highest l-weight representation if V = Uq ĝ · v for some highest l-weight vector v ∈ V . In [CP94], =-=[CP95a]-=-, it is shown that there is a bijection between the set of isomorphism classes of finite-dimensional irreducible highest l-weight Uqĝ-modules of type 1. Let L(m+) denote the irreducible highest l-wei... |

75 | Combinatorics of q-characters of finite-dimensional representations of quantum affine algebras
- Frenkel, E
(Show Context)
Citation Context ...os of the Drinfeld polynomials of the top module T . The q-character theory and the FM algorithm are important tools to study the representation theory of quantum affine algebras, see [NT98], [FR98], =-=[FM01]-=-. The main tools in this paper are the q-character theory and the FM algorithm. If the q-character of a module contains only one dominant monomial, then it is called a special module. Let g be the sim... |

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
(Show Context)
Citation Context ...Introduction The T-systems are some families of relations in the Grothendieck ring of the category of the finite-dimensional modules of quantum affine algebras (or Yangians), see [K83], [K84], [K87], =-=[KR90]-=-, [KNS94], [Nak03], [Her06]. The T-systems are widely applied to representation theory, combinatorics and integrable systems, see the recent survey [KNS11]. The modules in the usual T-systems are the ... |

71 | Cluster algebras and quantum affine algebras
- Hernandez, Leclerc
(Show Context)
Citation Context ...12], [CG11], [Her07], [LM12], [Mou10], [MF11], [MY12a], [MY12b], [MY12c], [Nao12]. The finite dimensional representations of Uqĝ and cluster algebras are closely related, see [IIKKN13a], [IIKKN13b], =-=[HL10]-=-, [HL13], [Nak11]. The relations in the extended Tsystems are special relations in the cluster algebras. A cluster algebra algorithm for computing q-characters of Kirillov-Reshetikhin modules for any ... |

64 | Representations of Yangians with Gelfand-Zetlin bases
- Nazarov
- 1998
(Show Context)
Citation Context ...the union of zeros of the Drinfeld polynomials of the top module T . The q-character theory and the FM algorithm are important tools to study the representation theory of quantum affine algebras, see =-=[NT98]-=-, [FR98], [FM01]. The main tools in this paper are the q-character theory and the FM algorithm. If the q-character of a module contains only one dominant monomial, then it is called a special module. ... |

62 | t-analogs of q-characters of Kirillov-Reshetikhin modules of quantum affine algebras, Represent. Theory 7
- Nakajima
- 2003
(Show Context)
Citation Context ...T-systems are some families of relations in the Grothendieck ring of the category of the finite-dimensional modules of quantum affine algebras (or Yangians), see [K83], [K84], [K87], [KR90], [KNS94], =-=[Nak03]-=-, [Her06]. The T-systems are widely applied to representation theory, combinatorics and integrable systems, see the recent survey [KNS11]. The modules in the usual T-systems are the Kirillov-Reshetikh... |

50 |
The Kirillov-Reshetikhin conjecture and solutions of T -systems
- Hernandez
(Show Context)
Citation Context ... are some families of relations in the Grothendieck ring of the category of the finite-dimensional modules of quantum affine algebras (or Yangians), see [K83], [K84], [K87], [KR90], [KNS94], [Nak03], =-=[Her06]-=-. The T-systems are widely applied to representation theory, combinatorics and integrable systems, see the recent survey [KNS11]. The modules in the usual T-systems are the Kirillov-Reshetikhin module... |

49 | Quiver varieties and cluster algebras
- Nakajima
(Show Context)
Citation Context ...r07], [LM12], [Mou10], [MF11], [MY12a], [MY12b], [MY12c], [Nao12]. The finite dimensional representations of Uqĝ and cluster algebras are closely related, see [IIKKN13a], [IIKKN13b], [HL10], [HL13], =-=[Nak11]-=-. The relations in the extended Tsystems are special relations in the cluster algebras. A cluster algebra algorithm for computing q-characters of Kirillov-Reshetikhin modules for any untwisted quantum... |

45 | Minimal affinizations of representations of quantum groups: the rank 2 case, preprint
- Chari
- 1994
(Show Context)
Citation Context ...ontains minimal affinizations of quantum affine algebras. The family of minimal affinizations is an important family of irreducible modules which contains the Kirillov-Reshetikhin modules, see [C95], =-=[CP95b]-=-, [CP96a], [CP96b]. The minimal affinizations are also interesting from the physical point of view, see Remark 4.2 of [FR92] and [C95]. Minimal affinizations are studied intensively in recently years,... |

27 | On minimal affinizations of representations of quantum groups - Hernandez |

26 | T-systems and Y-systems in integrable systems
- Kuniba, Nakanishi, et al.
(Show Context)
Citation Context ...ebras (or Yangians), see [K83], [K84], [K87], [KR90], [KNS94], [Nak03], [Her06]. The T-systems are widely applied to representation theory, combinatorics and integrable systems, see the recent survey =-=[KNS11]-=-. The modules in the usual T-systems are the Kirillov-Reshetikhin modules. The KirillovReshetikhin modules are simplest examples of irreducible finite-dimensional modules over quantum affine algebras.... |

23 |
Identities for the Rogers dilogarithm function connected with simple Lie algebras
- Kirillov
- 1989
(Show Context)
Citation Context ...ns. 1. Introduction The T-systems are some families of relations in the Grothendieck ring of the category of the finite-dimensional modules of quantum affine algebras (or Yangians), see [K83], [K84], =-=[K87]-=-, [KR90], [KNS94], [Nak03], [Her06]. The T-systems are widely applied to representation theory, combinatorics and integrable systems, see the recent survey [KNS11]. The modules in the usual T-systems ... |

22 |
Completeness of states of the generalized Heisenberg magnet
- Kirillov
- 1987
(Show Context)
Citation Context ...nizations. 1. Introduction The T-systems are some families of relations in the Grothendieck ring of the category of the finite-dimensional modules of quantum affine algebras (or Yangians), see [K83], =-=[K84]-=-, [K87], [KR90], [KNS94], [Nak03], [Her06]. The T-systems are widely applied to representation theory, combinatorics and integrable systems, see the recent survey [KNS11]. The modules in the usual T-s... |

15 |
Combinatorial identities and completeness of states of the Heisenberg magnet.
- Kirillov
- 1983
(Show Context)
Citation Context ...al affinizations. 1. Introduction The T-systems are some families of relations in the Grothendieck ring of the category of the finite-dimensional modules of quantum affine algebras (or Yangians), see =-=[K83]-=-, [K84], [K87], [KR90], [KNS94], [Nak03], [Her06]. The T-systems are widely applied to representation theory, combinatorics and integrable systems, see the recent survey [KNS11]. The modules in the us... |

12 | Restricted limits of minimal affinizations - Moura |

7 | Minimal affinizations as projective objects
- Chari, Greenstein
(Show Context)
Citation Context ...imal affinizations are also interesting from the physical point of view, see Remark 4.2 of [FR92] and [C95]. Minimal affinizations are studied intensively in recently years, see for example, [CMY12], =-=[CG11]-=-, [Her07], [LM12], [Mou10], [MF11], [MY12a], [MY12b], [MY12c], [Nao12]. The finite dimensional representations of Uqĝ and cluster algebras are closely related, see [IIKKN13a], [IIKKN13b], [HL10], [HL... |

5 |
A cluster algebra approach to q-characters of Kirillov-Reshetikhin modules (2013), arXiv :1303.0744
- Hernandez, Leclerc
(Show Context)
Citation Context ...11], [Her07], [LM12], [Mou10], [MF11], [MY12a], [MY12b], [MY12c], [Nao12]. The finite dimensional representations of Uqĝ and cluster algebras are closely related, see [IIKKN13a], [IIKKN13b], [HL10], =-=[HL13]-=-, [Nak11]. The relations in the extended Tsystems are special relations in the cluster algebras. A cluster algebra algorithm for computing q-characters of Kirillov-Reshetikhin modules for any untwiste... |

5 | Periodicities of T and Y-systems, dilogarithm identities, and cluster algebras I: Type Br
- Inoue, Iyama, et al.
(Show Context)
Citation Context ...s, see for example, [CMY12], [CG11], [Her07], [LM12], [Mou10], [MF11], [MY12a], [MY12b], [MY12c], [Nao12]. The finite dimensional representations of Uqĝ and cluster algebras are closely related, see =-=[IIKKN13a]-=-, [IIKKN13b], [HL10], [HL13], [Nak11]. The relations in the extended Tsystems are special relations in the cluster algebras. A cluster algebra algorithm for computing q-characters of Kirillov-Reshetik... |

5 | Demazure modules and graded limits of minimal affinizations ,Represent. Theory 17
- Naoi
- 2013
(Show Context)
Citation Context ...ew, see Remark 4.2 of [FR92] and [C95]. Minimal affinizations are studied intensively in recently years, see for example, [CMY12], [CG11], [Her07], [LM12], [Mou10], [MF11], [MY12a], [MY12b], [MY12c], =-=[Nao12]-=-. The finite dimensional representations of Uqĝ and cluster algebras are closely related, see [IIKKN13a], [IIKKN13b], [HL10], [HL13], [Nak11]. The relations in the extended Tsystems are special relat... |

4 | Graded limits of minimal affinizations and beyond: the multiplicity free case for type E6
- Moura, Pereira
(Show Context)
Citation Context ...sting from the physical point of view, see Remark 4.2 of [FR92] and [C95]. Minimal affinizations are studied intensively in recently years, see for example, [CMY12], [CG11], [Her07], [LM12], [Mou10], =-=[MF11]-=-, [MY12a], [MY12b], [MY12c], [Nao12]. The finite dimensional representations of Uqĝ and cluster algebras are closely related, see [IIKKN13a], [IIKKN13b], [HL10], [HL13], [Nak11]. The relations in the... |

4 | Affinization of category O for quantum groups - Mukhin, Young - 2013 |

3 | Prime representations from a homological perspective. arXiv:1112.6376
- Chari, Moura, et al.
- 1994
(Show Context)
Citation Context .... The minimal affinizations are also interesting from the physical point of view, see Remark 4.2 of [FR92] and [C95]. Minimal affinizations are studied intensively in recently years, see for example, =-=[CMY12]-=-, [CG11], [Her07], [LM12], [Mou10], [MF11], [MY12a], [MY12b], [MY12c], [Nao12]. The finite dimensional representations of Uqĝ and cluster algebras are closely related, see [IIKKN13a], [IIKKN13b], [HL... |

3 | Path description of type B q-characters
- Mukhin, Young
(Show Context)
Citation Context ...om the physical point of view, see Remark 4.2 of [FR92] and [C95]. Minimal affinizations are studied intensively in recently years, see for example, [CMY12], [CG11], [Her07], [LM12], [Mou10], [MF11], =-=[MY12a]-=-, [MY12b], [MY12c], [Nao12]. The finite dimensional representations of Uqĝ and cluster algebras are closely related, see [IIKKN13a], [IIKKN13b], [HL10], [HL13], [Nak11]. The relations in the extended... |

2 |
The q-characters of representations of quantum affine algebras and deformations of W-algebras, Recent developments in quantum affine algebras and related topics
- Reshetikin
- 1998
(Show Context)
Citation Context ...n of zeros of the Drinfeld polynomials of the top module T . The q-character theory and the FM algorithm are important tools to study the representation theory of quantum affine algebras, see [NT98], =-=[FR98]-=-, [FM01]. The main tools in this paper are the q-character theory and the FM algorithm. If the q-character of a module contains only one dominant monomial, then it is called a special module. Let g be... |

1 |
Minimal affnizations of representations of quantum groups: the rank 2
- Chari
- 1995
(Show Context)
Citation Context ...stems contains minimal affinizations of quantum affine algebras. The family of minimal affinizations is an important family of irreducible modules which contains the Kirillov-Reshetikhin modules, see =-=[C95]-=-, [CP95b], [CP96a], [CP96b]. The minimal affinizations are also interesting from the physical point of view, see Remark 4.2 of [FR92] and [C95]. Minimal affinizations are studied intensively in recent... |

1 |
Quantum a?ne algebras and holonomic difference equations
- Frenkel, Reshetikhin
- 1992
(Show Context)
Citation Context ...ucible modules which contains the Kirillov-Reshetikhin modules, see [C95], [CP95b], [CP96a], [CP96b]. The minimal affinizations are also interesting from the physical point of view, see Remark 4.2 of =-=[FR92]-=- and [C95]. Minimal affinizations are studied intensively in recently years, see for example, [CMY12], [CG11], [Her07], [LM12], [Mou10], [MF11], [MY12a], [MY12b], [MY12c], [Nao12]. The finite dimensio... |

1 | Extended T-system of type G2
- Li, Mukhin
(Show Context)
Citation Context ...les are simplest examples of irreducible finite-dimensional modules over quantum affine algebras. Recently, the usual T-systems have been generalized to the so called extended T-systems, see [MY12b], =-=[LM12]-=-. The extended T-systems contains minimal affinizations of quantum affine algebras. The family of minimal affinizations is an important family of irreducible modules which contains the Kirillov-Reshet... |

1 | Quantum loop algebras and l-root operators, arXiv:1206.6657, 1–27. School of mathematics and statistics - Young |