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

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.