Coloured Hopf algebras

Abstract

Quite recently, a ``coloured'' extension of the Yang-Baxter equation has appeared in the literature and various solutions of it have been proposed. In the present contribution, we introduce a generalization of Hopf algebras, to be referred to as coloured Hopf algebras, wherein the comultiplication, counit, and antipode maps are labelled by some colour parameters. The latter may take values in any finite, countably infinite, or uncountably infinite set. A straightforward extension of the quasitriangularity property involves a coloured universal ${\cal R}$-matrix, satisfying the coloured Yang-Baxter equation. We show how coloured Hopf algebras can be constructed from standard ones by using an algebra isomorphism group, called colour group. Finally, we present two examples of coloured quantum universal enveloping algebras.