Related papers: Extended T-systems
Associated to quantum affine general linear Lie superalgebras are two families of short exact sequences of representations whose first and third terms are irreducible: the Baxter TQ relations involving infinite-dimensional representations;…
We prove a family of 3-term relations in the Grothendieck ring of the category of finite-dimensional modules over the affine quantum algebra of type $G_2$ extending the celebrated $T$-system relations of type $G_2$. We show that these…
Frenkel-Reshetikhin introduced $q$-characters of finite dimensional representations of quantum affine algebras. We give a combinatorial algorithm to compute them for all simple modules. Our tool is $t$-analogue of the $q$-characters, which…
We propose the notion of q-characters for finite-dimensional representations of quantum affine algebras. It is motivated by our theory of deformed W-algebras. We show that the q-characters give rise to a homomorphism from the Grothendieck…
Generalized Baxter's TQ-relations and the QQ-system are systems of algebraic relations in the category O of representations of the Borel subalgebra of the quantum affine algebra U_q(g^), which we established in our earlier works…
We consider various specializations of the non-twisted quantum affine algebras at roots of unity. We define and study the q-characters of their finite-dimensional representations.
We develop the theory of $q$-characters for quantum affine superalgebras of type $A$ in connection with deformed Cartan matrices. To achieve this, we establish a Khoroshkin-Tolstoy-type multiplicative formula of the universal $R$-matrix of…
The extended $T$-systems are a number of short exact sequences in the category of finite-dimensional modules over the quantum affine algebras of types $A_n^{(1)}$ and $B_n^{(1)}$, introduced by Mukhin and Young as a generalization of the…
We continue the study of extended T-systems of quantum affine algebras. We find a sub-system of the extended T-system of the quantum affine algebra $U_q \hat{\mathfrak{g}}$ of type $C_3$. The sub-system consisting of four systems which are…
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 first review some invariant theoretic results about the finite subgroups of SU(2) in a quick algebraic way by using the McKay correspondence and quantum affine Cartan matrices. By the way it turns out that some parameters (a,b,h;p,q,r)…
We give a set of sufficient conditions for a Laurent polynomial to be the q-character of a finite-dimensional irreducible representation of a quantum affine group. We use this result to obtain an explicit path description of q-characters…
We study finite dimensional representations of the quantum affine algebra, using geometry of quiver varieties introduced by the author. As an application, we obtain character formulas expressed in terms of intersection cohomologies of…
The classification of irreducible, spherical characters of the infinite-dimensional unitary/orthogonal/symplectic groups can be obtained by finding all possible limits of normalized, irreducible characters of the corresponding…
We define the cluster algebra associated with the Q-system for the Kirillov-Reshetikhin characters of the quantum affine algebra $U_q(\hat{\g})$ for any simple Lie algebra g, generalizing the simply-laced case treated in [Kedem 2007]. We…
We use the $q$-characters to compute explicit expressions of the $R$-matrices for first fundamental representations of all types of twisted quantum affine algebras.
The T-systems and Y-systems are classes of algebraic relations originally associated with quantum affine algebras and Yangians. Recently they were generalized to quantum affinizations of quantum Kac-Moody algebras associated with a wide…
We classify the irreducible finite-dimensional representations of the twisted quantum affine algebras.
The q-characters were introduced by Frenkel and mReshetikhin to study finite dimensional representations of the untwisted quantum affine algebra for q generic. The $\epsilon$-characters at roots of unity were constructed by Frenkel and…
We develop a theory of sesquilinear forms over finite fields, investigating their representations via polynomials and coefficient matrices, along with classification results for these forms. Through their connection to quadratic forms, we…