On quantum shuffle and quantum affine algebras

Dual canonical bases, quantum shuffles and q-characters

Rosso and Green have shown how to embed the positive part Uq(n) of a quantum enveloping algebra Uq(g) in a quantum shuffle algebra. In this paper we study some properties of the image of the dual canonical basis B∗ of Uq(n) under this embedding Φ. This is motivated by the fact that when g is of type Ar, the elements of Φ(B∗) are q-analogues of irreducible characters of the affine Iwahori-Hecke ...

Reiter’s Properties for the Actions of Locally Compact Quantum Goups on von Neumann Algebras

Quantum Z-algebras and Representations of Quantum Affine Algebras

Generalizing our earlier work, we introduce the homogeneous quantum Z-algebras for all quantum affine algebras Uq(ĝ) of type one. With the new algebras we unite previously scattered realizations of quantum affine algebras in various cases. As a result we find a realization of Uq(F (1) 4 ). 0. Introduction In 1981 Lepowsky and Wilson introduced (principal) Z-algebras as a tool to construct expli...

Quantum vertex C((t))-algebras and quantum affine algebras

We give a summary of the theory of (weak) quantum vertex C((t))algebras and the association of quantum affine algebras with (weak) quantum vertex C((t))-algebras.

Quantum Toroidal Algebras and Their Representations

Quantum toroidal algebras (or double affine quantum algebras) are defined from quantum affine Kac-Moody algebras by using the Drinfeld quantum affinization process. They are quantum groups analogs of elliptic Cherednik algebras (elliptic double affine Hecke algebras) to whom they are related via Schur-Weyl duality. In this review paper, we give a glimpse on some aspects of their very rich repre...