FREE-FIELD REPRESENTATION OF GROUP ELEMENT FOR SIMPLE QUANTUM GROUPS

Abstract

A representation of the group element (also known as "universal [Formula: see text]-matrix") which satisfies Δ(g)=g⊗g, is given in the form [Formula: see text]where [Formula: see text], qi= q‖αi‖2/2 and Hi=2Hαi/ ‖αi‖2 and T±i are the generators of quantum group associated respectively with Cartan algebra and the simple roots. The "free fields" χ, ϕ, ψ form a Heisenberg-like algebra: [Formula: see text] We argue that the d G -parametric "manifold" which g spans in the operator-valued universal envelopping algebra, can also be invariant under the group multiplication g→ g′ · g′′. The universal ℛ-matrix with the property that ℛ(g⊗ I)(I⊗g)= (I⊗ g)(g⊗ I)ℛ is given by the usual formula [Formula: see text]