Let $\frak{g}$ be a complex semi-simple Lie algebra, and $\lambda,\mu \in P^+$ two positive dominant weights with corresponding irreducible representations $V(\lambda)$ and $V(\mu)$. The tensor product $V(\lambda) \otimes V(\mu)$ has the well-known decomposition into irreducible components given by the Littlewood-Richardson rule.

My question is whether or not this decomposition also exists in the quantum case, in other words, for the quantized enveloping algebra $U_q(\frak{g})$ with irreducible representations $V(\lambda)$ and $V(\mu)$, does $V(\lambda) \otimes V(\mu)$ have a "quantum Littlewood-Richardson" decomposition?