Let $\mathfrak g$ be a semi-simple Lie algebra (We can assume $\mathfrak g=sl(n)$ for simplicity) and let $O_q(\mathfrak g)$ be the corresponding quantum algebra of functions. Then $O_q(\mathfrak g)^{\otimes N}$ is a left $O_q(\mathfrak g)$-comodule or, dually, a right $U_q(\mathfrak g)$-module. Let $V_N=(O_q(\mathfrak g)^{\otimes N})^{U_q(\mathfrak g)}$ be its subspace invariant under the $U_q(\mathfrak g)$-action.
Did someone work out its Lusztig canonical basis?
Also, since Lusztig canonical bases are complicated, I wonder if other "nice" bases of it were studied perhaps?
