We study the tensor product decomposition of the split real quantum group \({\mathcal {U}}_{q\widetilde{q}}(\mathfrak {sl}(2,\mathbb {R}))\) from the perspective of finite-dimensional representation theory of compact quantum groups. It is known that the class of positive representations of \({\mathcal {U}}_{q\widetilde{q}}(\mathfrak {sl}(2,\mathbb {R}))\) is closed under taking tensor product. In this paper, we show that one can derive the corresponding Hilbert space decomposition, given explicitly by quantum dilogarithm transformations, from the Clebsch–Gordan coefficients of the tensor product decomposition of finite-dimensional representations of the compact quantum group \({\mathcal {U}}_q(\mathfrak {sl}_2)\) by solving certain functional equations arising from analytic continuation and using normalization arising from tensor products of canonical basis.

我们研究了分裂真实量子组的张量积分解\（{\ mathcal【U}} _ {Q \ widetilde {Q}}（\ mathfrak {SL}（2，\ mathbb {R}））\）从量子团的有限维表示理论的观点 已知\（{\ mathcal {U}} _ {q \ widetilde {q}}（\ mathfrak {sl}（2，\ mathbb {R}））\）的正表示形式是闭合的。张量积。在本文中，我们表明，可以从紧凑的量子组\（{\ mathcal { U}} _ q（\ mathfrak {sl} _2）\） 通过求解解析连续性产生的某些函数方程，并使用规范基础的张量积引起的归一化。