On Hopf algebraic structures of quantum toroidal algebras

Abstract

We define an algebra $\mathcal{U}_0$ using a simplified set of generators for the quantum toroidal algebra $U_q(sl_{n+1}, tor)$ and show that there exists an epimorphism from $\mathcal{U}_0$ to $U_q(sl_{n+1}, tor)$. We derive a closed formula of the comultiplication on the generators of $\mathcal{U}_0$ that extends that of the quantum affine algebra $U_q(\hat{sl}_{n+1})$. As a consequence, we show that $\mathcal{U}_0$ is a Hopf algebra for $n=1, 2$ and give conjectural formulas in the general case. We further show that $\mathcal{U}_0$ is isomorphic to a double algebra.