Department of Mathematics - Seminar on Pure Mathematics - Presentations of Quantum Affine Algebras and Extended Quantum Affine Algebras

Quantum affine algebras are quantum enveloping algebras of affine Lie algebras, introduced independently by Drinfeld and Jimbo in their study of the Yang-Baxter equation. Representation theory of quantum affine algebras depends mostly on the Drinfeld realization, which was announced in 1987. Ding and Frenkel proved the presentation using the Faddeev school's R-matrix method in type A in 1993. We will discuss our recent joint work with A. Molev and M. Liu on the Drinfeld realization for other classical types. Our proof extended the R-matrix method with the help of quasideterminants, and we constructed a sequence of larger quantum algebras associated with the spectral parameter dependent R-matrix. In the end, I will also briefly discuss another presentation of quantum extended affine algebras (jointly with F. Chen, F. Kong and S. Tan).