Generators of the Quantum Coordinate Algebras and Quantized Enveloping Algebra Representations

Question

The representations of the quantized enveloping algebra $U_q(\frak{sl}_n)$ are labeled by the positive weight lattice $P^+$ of the classical Lie algebra $\frak{sl}_n$. Moreover, the dual Hopf algebra $SL_q(N)$ is defined as the subspace of the Hopf dual of $U_q(\frak{sl}_n)$ consisting of the coordinate functions of these representations, where for $\lambda \in P^+$, $V(\lambda)$ the corresponding representation, $v \in V(\lambda)$, and $f \in V(\lambda)^*$, we define $$ C_{f,v}: U_q({\frak sl}_N) \to k, ~~~~~~~~ A \mapsto f(Av). $$ Now for $U_q(\frak{sl}_2)$, we have that $SL_q(2)$ is generated as an algebra by the coordinate algebra of the representation corresponding to the classical natural representation. Does this result extend to the general $U_q({\frak sl}_N)$ and $SL_q(N)$ case? In other words, is it correct to assume that the generators $u^i_j$, for the coordinate algebra of this representations are the same as the standard representations of $SL_q(N)$ Or more generally still, does it extend to all $U_q(\frak{g})$?

Consider now the representation coming from the exterior product of the natural representation with itself. I would like to know how the elements of the coordinate algebra of this representation look like in terms of the generators $u^i_j$. I can't see an easy guess that matches the dimensions.

Answer 1

For your first question, the answer is yes, as Casteels pointed out in the comments. The reason is that, for $\mathfrak{sl}_N$, every finite-dimensional irreducible representation appears as a subrepresentation of some tensor power of the natural representation. Since the finite-dimensional representation theory of the quantized enveloping algebra behaves the same as in the classical case (i.e. tensor products decompose in the same way), the same is true in the quantum setting.

I don't know off the top of my head an expression in the generators corresponding to the second exterior power of the natural representation. But you can find one as follows: take $V$ to be the natural representation of $U_q(\mathfrak{sl}_N)$, and decompose $V \otimes V$ explicitly into irreducible components. There will be two: one of them will correspond to $\mathrm{Sym}^2(V)$ and the other to $\Lambda^2(V)$. The generators $u_j^i$ are the coordinate functions for $V$, so this decomposition will give you expressions for the coordinate functions of the two subrepresentations of $V\otimes V$ in terms of the $u_j^i$.