Quantum Affine Algebras and their Representations

Abstract

We prove a highest weight theorem classifying irerducible finite--dimensional representations of quantum affine algebras and survey what is currently known about the structure of these representations.

19 pages

Cited by in corpus (42)

• Combinatorics of q-characters of finite-dimensional representations of quantum affine algebras
• The q-characters of representations of quantum affine algebras and deformations of W-algebras
• Existence of Kirillov-Reshetikhin crystals for nonexceptional types
• Kirillov--Reshetikhin crystals for nonexceptional types
• On minimal affinizations of representations of quantum groups
• Cylindric versions of specialised Macdonald functions and a deformed Verlinde algebra
• Minimal Affinizations of Representations of Quantum Groups: the simply--laced case
• A Quantum Affine Algebra for the Deformed Hubbard Chain
• X=M for symmetric powers
• Demazure structure inside Kirillov-Reshetikhin crystals
• Path description of type B q-characters
• Representations of Quantum Affine Superalgebras
• Geometric construction of representations of affine algebras
• Paths, tableaux, and q-characters of quantum affine algebras: the C_n case
• A Double Hall Algebra Approach to Affine Quantum Schur--Weyl Theory
• Symmetric quiver Hecke algebras and R-matrices of quantum affine algebras II
• Construction of perfect crystals conjecturally corresponding to Kirillov-Reshetikhin modules over twisted quantum affine algebras
• Combinatorial structure of Kirillov-Reshetikhin crystals of type D_n(1), B_n(1), A_{2n-1}(2)
• Quantum Grothendieck ring isomorphisms, cluster algebras and Kazhdan-Lusztig algorithm
• Finite-Dimensional Crystals B^{2,s} for Quantum Affine Algebras of type D_{n}^{(1)}
• Polynomial Relations for q-Characters via the ODE/IM Correspondence
• Graded Limits of Minimal Affinizations in Type D
• Affine highest weight categories and quantum affine Schur-Weyl duality of Dynkin quiver types
• Realization of affine type A Kirillov-Reshetikhin crystals via polytopes
• A crystal to rigged configuration bijection and the filling map for type $D_4^{(3)}$
• Rigged configuration bijection and proof of the $X=M$ conjecture for nonexceptional affine types
• M-systems and Cluster algebras
• On higher level Kirillov--Reshetikhin crystals, Demazure crystals, and related uniform models
• Extended $T$-System of Type $G_2$
• T-systems and Y-systems for quantum affinizations of quantum Kac-Moody algebras
• Positivity and periodicity of $Q$-systems in the WZW fusion ring
• Applications of Crystal Bases to Current Problems in Representation Theory
• Path description for $q$-characters of fundamental modules in type $C$
• A combinatorial realization of Kirillov-Reshetikhin crystals for type E arising from translations
• Positive level, negative level and level zero
• Quantum affine algebras and Grassmannians
• Raising/lowering maps and modules for the quantum affine algebra $U_q(s{l}_{2}hat)$
• Equivariant multiplicities via representations of quantum affine algebras
• Tensor products of level zero representations
• Irreducible Modules for the Quantum Affine Algebra $U_q(g)$ and its Borel subalgebra $U_q(g{)}^{\ge 0}$
• Minimal Affinizations of Representations of quantum groups: the U_q(g)--module structure
• Tensor Product of the Fundamental Representations for the Quantum Loop Algebras of Type A at Roots of Unity