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 $U_q(s{l}_{n+1},tor)$ and show that there exists an epimorphism from ${\mathcal{U}}_0$ to $U_q(s{l}_{n+1},tor)$. We derive a closed formula of the comultiplication on the generators of ${\mathcal{U}}_0$ that extends that of the quantum affine algebra $U_q({\widehat{sl}}_{n+1})$. 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.

26 pages, 1 figure; updated version

References in corpus (3)

• Coproduct for the Yangian of an affine Kac-Moody algebra
• Two-parameter Quantum Affine Algebra $U_{r,s}(\widehat{{\mathfrak{sl}}_n})$, Drinfel'd Realization and Quantum Affine Lyndon Basis
• Two-parameter twisted quantum affine algebras

Cited by in corpus (1)

• Two-parameter twisted quantum affine algebras