Quantum groups of Lie colour algebras fulfilling Cartan-Weyl paradigm

Abstract

Let $Γ$ be an additive abelian group equipped with a commutative factor $ω$. We describe the simple Lie colour algebras and the associated untwisted affine Lie colour algebras graded by $Γ$, 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 $Γ$. 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.