An explicit description of the fundamental unitary for SU(2) _q

Abstract

We give a concrete description of an isometryv from ℓ²(ℕ×ℤ×ℤ×ℤ) to ℓ²(ℕ×ℤ×ℕ×ℤ) whose existence has recently been discovered by Woronowicz [11]. The isometryv gives the comultiplication δ on the C^*-algebraA of the quantum group SU(2) _q through the formula δ(x)=v(x⊗1)v ^* (x∈A), where 1 is the identity operator on ℓ²(ℤ×ℤ). The matrix entries ofv are described in terms of littleq-Jacobi polynomials. Usingv, we give a concrete description of a unitary operatorV onH _η ⊗H _η such that (π_η⊗π_η)δ(x)=V(π_η(x⊗1)V ^*, whereH _η=ℓ²(ℕ×ℤ×ℕ) and π_η:A→L(H _η) is the GNS representation associated with the Haar state η onA. The operatorV satisfies the pentagonal identity of Baaj and Skandalis [1].