Abstracts

The universal R-matrix is an element of a completion of quantum groups, which endows its module category a structure of a braided monoidal category. In this talk, we prove the uniqueness and existence of the universal R-matrix, its multiplicative formula, and discuss its applications to canonical bases.

First, we will construct dual canonical bases of symmetric and exterior powers of the natural representations of U_n=U_q(gl_n). And then, we will derive dual canonical bases of the irreducible polynomial representation of U_n by realizing the representation by means of a homomorphism between symmetric and exterior powers.

For a given vertex algebra, one can associate an associative algebra called a twisted Zhu algebra provided a proper gradation on the vertex algebra. As one of the most important example, a finite W-algebra is a twisted Zhu algebra of an affine W-algebra. In this talk, I will explain twisted Zhu algebras by showing some examples, including finite W-algebras, and review some representation theory behind them.

강연 자료

In this talk, we review the relationship between RTT formalism and Drinfeld (current) presentation of the Yangian of type A. Using this connection, we interpret the isomorphism, introduced by Brundan-Kleshchev, from the Yangian of type A to finite rectangular W-algebra of type A. Also, we briefly review the homomorphism, constructed by Ueda, from the affine yangian to the universal enveloping algebra of the rectangular affine W-algebra and explain the relation between Brundan-Kleshchev’s map and Ueda’s map.

강연 자료

Dirac reduction is the way to deal with Hamiltonian mechanics with constraints. In this talk, I explain the structures of classical W-algebras in terms of the modified Dirac reduction and why this modification is needed. If time allows, I will tell you that the results in our work which are extended to W-superalgebra with (or without) supersymmetric context. This is the joint work with Arim Song and Uhirinn Suh.

강연 자료

Let g be a finite-dimensional simple Lie algebra over C. When g is of simply-laced type, a t-deformation K_t(g) of the Grothendieck ring of a category of finite-dimensional modules over a quantum loop algebra U_q(Lg) has a rich structure, especially (I) a deep connection with the dual canonical basis of the quantum group U_q(g) and (II) quantum cluster algebra structure of skew-symmetric type. In this talk, I briefly introduce the quantum virtual Grothendieck ring K_q(g), which can be viewed as a non-trivial generalization of K_t(g) for simply-laced types. Focused on the viewpoint of (I), I mainly explain some bases of K_q(g). Those bases are closely related to the dual-canonical/upper-global basis of finite type. This talk is based on joint work with Kyu-Hwan Lee and Se-jin Oh.

The Jantzen sum formula played an important role in calculating the decomposition number of symmetric groups. James and Mathas developed a q-analogue of Jantzen's sum formula and used it to characterize irreducible Weyl modules. In this talk, we will study the q-analogue of Jantzen's sum formula based on Mathas' book "Iwahori-Hecke algebras and Schur algebras of the symmetric group".

강연 자료

In this talk, I explain the blocks of Hecke algebra and q-Schur algebra, which is a building block of their module category. In addition, I'll give some results on the irreducibility of well-known modules, such as Weyl modules and Specht modules. This is a summary of Chapter 5 of the main reference: Andrew Mathas, Iwahori-Hecke algebras and Schur algebras of symmetric groups (1999).

강연 자료

Deformed W-algebras have been studied by many mathematicians because of their interesting relation to integrable models associated to quantum affine algebras. In this talk, I will explain the construction of deformed W-algebra suggested by E.Frenkel, N.Reshetikhin, M.A. Semenov-Tian-Shansky, which is an analogue of Drinfeld-Sokolov reduction for non-deformed W-algebras. If time permits, I will also explain its relation to quantum affine algebras by describing deformed Miura transformation.

강연 자료

The q-character is one of the most significant tools for finite-dimensional representations of quantum affine algebras. On one hand, one can regard the q-character as a kind of formal character and study it. On the other hand, the construction of it is largely motivated from a method of integrable systems, which is designed to solve the spectral problem of the Hamiltonian. The goal of this talk is to understand in this vein the corresponding (1+1)D quantum integrable model in terms of representations of quantum affine algebra. As always the key role is played by the R-matrix, which will be demonstrated in the toy example: spin 1/2 XXX chain.