Are there some references about the result Lemma 1.2 in BRAIDED SYMMETRIC AND EXTERIOR ALGEBRAS by Arkady Berenstein and Sebastian Zwicknagl?
I type Lemma 1.2 in the following.
Let $C \in Z(\widehat{U_q(g)})$ be the quantum Casimir element which acts on any simple finite dimentional $U_q(g)$-module $V_{\lambda}$ by scalar multiple $q^{(\lambda| 2\rho)}$, where $2\rho$ is the sum of all positive roots.
Let $R \in U_q(g) \widehat{\otimes} U_q(g)$ be the universal R-matrix and define $\mathcal{R}_{U, V}(u \otimes v)=\tau R(u \otimes v)$, $U, V$ are finite dimensional simple $U_q(g)$-modules. Then $\mathcal{R}^2 = \Delta(C^{-1}) \circ (C \otimes C)$. In particular, the restriction of $\mathcal{R}^2$ to the isotypic component $I_{\lambda,\mu}^{\nu}$ of $V_{\lambda} \otimes V_{\mu}$ is scalar multiplication by $q^{(\lambda|\lambda)+(\mu|\mu)-(\nu|\nu)+(2\rho|\lambda + \mu-\nu)}$.
Why $\mathcal{R}^2 = \Delta(C^{-1}) \circ (C \otimes C)$ and the restriction of $\mathcal{R}^2$ to the isotypic component $I_{\lambda,\mu}^{\nu}$ of $V_{\lambda} \otimes V_{\mu}$ is scalar multiplication by $q^{(\lambda|\lambda)+(\mu|\mu)-(\nu|\nu)+(2\rho|\lambda + \mu-\nu)}$?
Thank you very much.
