Quantum toroidal and shuffle algebras

Abstract

In this paper, we prove that the quantum toroidal algebra of gl_n is isomorphic to the double shuffle algebra of Feigin and Odesskii for the cyclic quiver. The shuffle algebra viewpoint will allow us to prove a factorization formula for the universal R-matrix of the quantum toroidal algebra.

arXiv commentThe previous version of this paper was broken into two parts: the present version contains the representation-theoretic half (to which we added a number of additional results) and the geometric half has been moved to arXiv:1811.01011