Related papers: Quantum Z-algebras and representations of quantum …

The concept of a quantum algebra is made easy through the investigation of the prototype algebras $u_{qp}(2)$, $su_q(2)$ and $u_{qp}(1,1)$. The latter quantum algebras are introduced as deformations of the corresponding Lie algebras~; this…

We describe a connection between finite--dimensional representations of quantum affine algebras and affine Hecke algebras.

We give explicit constructions of quantum symplectic affine algebras at level 1 using vertex operators.

We study the structure and representations of a family of vertex algebras obtained from affine superalgebras by quantum reduction. As an application, we obtain in a unified way free field realizations and determinant formulas for all…

We construct quantized versions of generic bases in quantum cluster algebras of finite and affine types. Under the specialization of $q$ and coefficients to 1, these bases are generic bases of finite and affine cluster algebras.

Lie groups and quantum algebras are connected through their common universal enveloping algebra. The adjoint action of Lie group on its algebra is naturally extended to related q-algebra and q-coalgebra. In such a way, quantum structure can…

The so called quantized algebras of functions on affine Hecke algebras of type A and the corresponding q-Schur algebras are defined and their irreducible unitarizable representations are classified.

We show that the quantum affine algebra U_{q}(A_{1}^{(1)}) and the quantum affine superalgebra U_{q}(C(2)^{(2)}) admit unified description. The difference between them consists in the phase factor which is equal to 1 for U_{q}(A_{1}^{(1)})…

We give a realization of crystal graphs for basic representations of the quantum affine algebra $U_q(C_2^{(1)})$ in terms of new combinatorial objects called the Young walls.

Quadratic algebras related to the reflection equations are introduced. They are quantum group comodule algebras. The quantum group $F_q(GL(2))$ is taken as the example. The properties of the algebras (center, representations, realizations,…

We describe the underlying U_q(g)--module structure of representations of quantum affine algebras.

We show that the quantum affine algebra $U_{q}(A_{1}^{(1)})$ and the quantum affine superalgebra $U_{q}(C(2)^{(2)})$ admit a unified description. The difference between them consists in the phase factor which is equal to 1 for…

We give a simplified description of quantum affine algebras in their loop presentation. This description is related to Drinfeld's new realization via halves of vertex operators. We also define an idempotent version of the quantum affine…

We classify the irreducible finite-dimensional representations of the twisted quantum affine algebras.

We construct level one representations of the quantum affine algebra $U_q(G_2^{(1)})$ by vertex operators from bosonic fields.

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.

Quantum algebras (also called quantum groups) are deformed versions of the usual Lie algebras, to which they reduce when the deformation parameter q is set equal to unity. From the mathematical point of view they are Hopf algebras. Their…

We produce 2-representations of the positive part of affine quantum enveloping algebras on their finite-dimensional counterparts in type $A_n$. These 2-representations naturally extend the right-multiplication 2-representation of…

In this lecture, we survey a number of recent results and developments regarding the representation theory of infinite-dimensional quantum groups (quantum affine algebras and related algebras), as well as their connections with cluster…

We give closed formulae for the q-characters of the fundamental representations of the quantum loop algebra of a classical Lie algebra in terms of a family of partitions satisfying some simple properties. We also give the multiplicities of…