I am trying to learn some basic theory of quantum groups $U_q(\mathfrak{g})$, where $\mathfrak{g}$ is a simple Lie algebra, say $sl_n(\mathbb{C})$. As far as I heard the finite dimensional representation theory of them is well understood.
I would be interested to see examples of representations of $U_q(\mathfrak{g})$ which come not from the problem of classification of representations, but rather are either "natural" or typical from the point of view of quantum groups, or appear in applications or situations unrelated to the classification problem.
To illustrate what I mean let me give examples of each kind of represenations of Lie algebras since this situation is more familiar to me.
1) Let $\mathfrak{g}$ be a classical complex Lie algebra such as $sl_n,so(n), sp(2n)$. Then one has the standard representation of it, its dual, and tensor products of arbitrary tensor powers of them.
2) Example of very different nature comes from complex geometry. The complex Lie algebra $sl_2$ acts on the cohomology of any compact Kahler manifold (hard Lefschetz theorem). Analogously $so(5)$ acts on the cohomology of any compact hyperKahler manifold (this was shown by M. Verbitsky).
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If i have correctly understood your question, there are various such examples, arising from mathematical physics contexts (where some of the original motivations for the study of quantum groups first appeared).
One such example, is the case of the $q$-Heisenberg algebra: If we consider the (usual) 3d Heisenberg Lie algebra $L_H$, generated by $a,a^{\dagger},H$ subject to the relations: $$ [a,a^\dagger]=H, \ \ \ \ \ \ \ \ \ [H,a]=[H,a^\dagger]=0 $$ then the $q$-deformed Heisenberg algebra (with $q$ a non-zero parameter), may be defined in terms of generators $a,a^\dagger,q^\frac{H}{2},q^{-\frac{H}{2}}$ and $1$ and relations: $$ q^{\pm\frac{H}{2}}q^{\mp\frac{H}{2}}=1, \ \ \ \ \ [q^\frac{H}{2},a]=[q^\frac{H}{2},a^\dagger]=0, \ \ \ \ \ [a,a^\dagger]=\frac{q^H-q^{-H}}{q-q^{-1}} $$ (Of course now $[.,.]$ is no more the Lie bracket but simply the usual commutator). This is known to be a quasitriangular hopf algebra. It may be thought of, as the $q$-deformation $U_q(L_H)$ of the universal enveloping algebra $U(L_H)$ of the Heisenberg Lie algebra $L_H$.
It can be shown, that, if $q$ is a real number, then the unitary representations of $U_q(L_H)$ are parameterized by a real, positive parameter $\hbar$. If we denote the basis vectors by $$H_\hbar=\{|n,\hbar\rangle\big{|}n=0,1,2,...\}$$ then the action of the generators is given by: $$ |n,\hbar\rangle=\frac{(a^\dagger)^n}{[\hbar]^{\frac{n}{2}}\sqrt{n!}}|0,\hbar\rangle, \ \ \ \ \ q^{\pm\frac{H}{2}}|0,\hbar\rangle=q^{\pm\frac{\hbar}{2}}|0,\hbar\rangle, \ \ \ \ \ a|0,\hbar\rangle=0 $$ where $[\hbar]=\frac{q^\hbar-q^{-\hbar}}{q-q^{-1}}$. This a deformation of the usual Fock representation of the Heisenberg Lie algebra. If you are interested in similar examples, you can find more in S. Majid's book, "Foundations of Quantum group theory".
Furthermore, various $q$-deformations of the harmonic oscillator algebra can be used for a more systematic way of constructing such examples: Although most of $q$-deformed CCR are not quantum groups themselves (up to my knowledge there are generally no known hopf algebra structures for such algebras), suitable homomorphisms from quantum groups $U_q(g)$ (with $g$ being any Lie (super)algebra) to $q$-deformations of the harmonic oscillator can be used (such homomorphisms are usually called "realizations" in the literature) to pull back the $q$-deformed fock spaces of the $q$-deformed oscillators to representations of the corresponding quantum group $U_q(g)$.
Such methods have been applied since the '80's: L. C. Biedenharn and A. J. Macfarlane have provided descriptions of $su_q(2)$ deformations and their corresponding representations. A more complete account can be found in: Quantum Group Symmetry and Q-tensor Algebras. Similar methods have been used for the study of $su_q(1,1)$ deformed lie algebra representations. Two parameter $(q,s)$-deformations have also been studied, see for example the case of $sl_{q,s}(2)$ ... etc. These methods have also been extended to the case of deformations of the universal enveloping algebras of Lie superalgebras as well. See for example this article or this one.
The mathematical physics literature abounds of such examples during the last couple of decades.
The situation is similar to the way, various bosonic or fermionic realizations of Lie (super)algebras have been used to construct Lie (super)algebra representations, initiating from the usual symmetric/antisymmetric bosonic/fermionic Fock spaces: Such are the Holstein-Primakoff, the Dyson or the Schwinger (see: ch.3.8) realizations (see also: Dictionary on Lie Algebras and Superalgebras). If you are interested in such topics (in the deformed or the undeformed sense), i can provide further references (i have done some work on similar stuff during my phd thesis).