Macdonald's Symmetric Polynomials as Zonal Spherical Functions on Some Quantum Homogeneous Spaces

q-deformed Jacobi identity, q-oscillators and q-deformed infinite-dimensional algebras

Matrix factorizations and representations of quivers II: Type ADE case

This is a review paper on the Gelfand-Tsetlin type bases for representations of the classical Lie algebras. Different approaches to construct the original Gelfand-Tsetlin bases for representations of the general linear Lie algebra are discussed. Weight basis constructions for representations of the orthogonal and symplectic Lie algebras are reviewed. These rely on the representation theory of the B,C,D type twisted Yangians

Gelfand–Tsetlin Bases for Classical Lie Algebras

We study a triangulated category of graded matrix factorizations for a polynomial of type ADE. We show that it is equivalent to the derived category of finitely generated modules over the path algebra of the corresponding Dynkin quiver. Also, we discuss a special stability condition for the triangulated category in the sense of T. Bridgeland, which is naturally defined by the grading.

Matrix Factorizations and Representations of Quivers II: type ADE case

We give a presentation of an elliptic Weyl group W(R) (=the Weyl group for an elliptic root system*0 R) in terms of the elliptic Dynkin diagram F(R, G) for the elliptic root system. The presentation is a generalization of a Coxeter system: the generators are in one to one correspondence with the vertices of the diagram and the relations consist of two groups : i) elliptic Coxeter relations attached to the diagram, and ii) a flniteness condition on the Coxeter transformation attached to the diagram. The group defined only by the elliptic Coxeter relations is isomorphic to the central extension W(R, G) of W(R) by an infinite cyclic group, called the hyperbolic extension of W(R). *) an elliptic root system=a 2-extended affine root system (see the introduction and the remark at its end).

https://www.ems-ph.org/journals/show_pdf.php?issn=0034-5318&vol=33&iss=2&rank=7

Extended affine root systems III (elliptic Weyl groups)

The Gelfand-Zetlin basis of Uq(gl(N+1)) modules is constructed via the lowering operator method.

Gelfand-Zetlin basis for Uq(gl(N+1)) modules

The Gelfand-Tsetlin basis for irreducible Uq(gl(N+1))-modules of finite dimensions is constructed by means of the lowering operators

Construction of Gelfand-Tsetlin basis for U q gl N+1 -modules

We will study zonal spherical functions on quantum symmetric space GL q (N+1)/O q (N+1)), and will show that those for the case N=2 are given by Macdonald's polynomials of the A2 type. Some q-analogues of hypergeometric series associated with the quantum symmetric spaces will be discussed.

Tadayoshi Takebayashi

Papers

Extended affine root systems III (elliptic Weyl groups)

Gelfand-Zetlin basis for Uq(gl(N+1)) modules

Construction of Gelfand-Tsetlin basis for U q gl N+1 -modules

Zonal spherical functions on quantum symmetric spaces and MacDonald's symmetric polynomials

Defining Relations of Weyl Groups for Extended Affine Root Systems A(1,1)l, B(1,1)l, C(1,1)l, D(1,1)l