On Hopf algebraic structures of quantum toroidal algebras

We define an algebra 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 u(0) to U-q(sl(n+1),tor). We derive a closed formula of the comultiplication on the generators of u(0) that extends that of the quantum affine algebra U-q((sl) over cap (n+1)). As a consequence, we show that u(0) is a Hopf algebra for n = 1, 2 and give conjectural formulas in the general case. We further show that u(0) is isomorphic to a double algebra.