Quantum groups of Lie colour algebras fulfilling Cartan-Weyl paradigm

Abstract

Let $\Gamma$ be an additive abelian group equipped with a commutative factor $\omega$. We describe the simple Lie colour algebras and the associated untwisted affine Lie colour algebras graded by $\Gamma$, which fulfil the Cartan-Weyl paradigm. The quantised universal enveloping algebras of these (affine) Lie colour algebras are constructed, which are colour analogues of the Drinfeld-Jimbo quantum groups including the latter as the special case of trivial $\Gamma$. We develop the quasi-triangular Hopf colour algebraic structure of these ``colour quantum groups'', which has immediate applications in areas such as knot theory and statistical mechanics.