On explicit realization of algebra of complex divided powers of $U_{q}(\mathfrak{sl}(2))$

TL;DR

This paper provides an explicit realization of complex powers of generators in the quantum group $U_q(sl(2))$, confirming their algebraic relations and connecting them to quantum dilogarithm identities.

Contribution

It introduces a concrete realization of complex powers in $U_q(sl(2))$ that satisfies all algebraic relations, including a new integral identity involving quantum dilogarithm.

Findings

Verification of commutation relations for complex powers

Proof of generalized Kac's identity in this realization

Equivalence to a quantum dilogarithm integral identity

Abstract

In this note we prove that the explicit realization of arbitrary complex powers of generators of quantum group $U_{q}(\mathfrak{sl}(2))$ satisfies all the commutation relations of the algebra of complex powers, including the generalized Kac's identity which was announced in our previous paper. It turns out that the latter identity in this realization is equivalent to $6-9$ integral identity on quantum dilogarithm.