# MS Seminar (Mathematics - String Theory)

Speaker: Yoshihisa Saito (U Tokyo) On quantized elliptic algebras Tue, Feb 21, 2017, 15:30 - 17:00 Seminar Room B Let $\mathfrak{g}$ be a semisimple Lie algebra, and $\mathfrak{g}_{tor}$ the corresponding toridal Lie algebra. Namely, $\mathfrak{g}_{tor}$ is the universal central extension of double loop Lie algebra $\mathfrak{g}\otimes\mathbb{C}[s^{\pm 1},t^{\pm 1}]$.Around 2000, Kyoji Saito and Yoshii gave a new presentation of $\mathfrak{g}_{tor}$ by finitely many generators and relations, attached with elliptic Dynkin diagrams. In this talk, we introduce a q-analogue of K. Saito and Yoshii's presentation, so-called quantum elliptic algebras''. In addition, we show that these algebras are isomorphic to quantum toroidal algebras for several types. In other words, we give a presentation of quantum toroidal algebras by finitely many generators and relations.