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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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    $\begingroup$ Have you looked into the papers (and possibly the book) by Lusztig? Keep in mind that there are two very different possibilities: 1) the parameter often called $q$ or $v$ may be an arbitrary complex number not a root of unity (or even an indeterminate), or 2) it may be a root of unity. The latter case is least well understood, and often resembles the theory in prime characteristic. The former case is closer to classical representation theory. Anyway, Lusztig (and Andersen et al.) do give some examples. Jantzen's AMS book is very useful. $\endgroup$ – Jim Humphreys Oct 6 '17 at 0:34
  • $\begingroup$ @JimHumphreys: Lusztig's book looks to me quite technical. If one could show a specific place with relevant examples, that would be helpful. $\endgroup$ – MKO Oct 6 '17 at 7:12
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    $\begingroup$ Probably you'd get more insight from Chapters 2 and 5 of Jantzen's 1996 AMS text Lectures on Quantum Groups, even though the notation gets heavy as in Lusztig's book. Jantzen does attempt to provide more of a textbook approach, though it isn't easy in this subject. $\endgroup$ – Jim Humphreys Oct 6 '17 at 11:25
  • $\begingroup$ In the case of a quantum group arising from the rank 1 simple Lie algebra, a classic paper by Reshetikhin and Turaev may already be familiar to you: mathscinet.ams.org/mathscinet-getitem?mr=1091619 $\endgroup$ – Jim Humphreys Jan 6 '18 at 21:58
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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).

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    $\begingroup$ This is valuable, but I think the question involves a more specific class of "quantum groups" (or quantized enveloping algebras) coming from simple Lie algebras. The subject spreads out quite a bit. $\endgroup$ – Jim Humphreys Jan 6 '18 at 21:50
  • $\begingroup$ Prof. Humphreys, thank you for your feedback. You are probably right in your point that the example spreads out the focus of the OP. However, i think that the techniques mentioned in the last paragraph, utilizing Lie (super)algebra realizations, might be of some interest for the quantized universal enveloping algebras coming from simple Lie algebras as well. $\endgroup$ – Konstantinos Kanakoglou Jan 6 '18 at 22:06
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The noncommutative complex geometry of the quantum flag manifolds provides a rich family of examples of representations of quantized enveloping algebras. For example, there are a number of papers on $q$-deformations of the Borel-Weil theorem, which realises Lie algebra representations on the holomorphic sections of line bundles. One series of papers starts with Majid's paper on the noncommutative spin geometry of the Podles sphere (quantum $\mathbb{CP}^1$, where Borel-Weil is discussed in the final section. A second paper on Borel-Weil for the Podles sphere was later written by Khalkhali, Landi, and van Suijlekom. Moreover, this result was then extended to all quantum projective space by Khalkhali and Moatadelro here. Finally, the extension to all the quantum Grassmannians is given in this paper. A google search will give other approaches to this question.

The representation of $\frak{sl}_2$ on the de Rham complex of a Kahler manifold also has a direct noncommutative generalisation. What one needs is a differential calculus endowed with a noncommutative Kahler structure. (This is basically a noncommutative complex structure on the calculus together with a $(1,1)$-form whose Lefschetz map $L$ behaves like a classical Lefschetz map.) The map $L$ admits an adjoint $\Lambda$, the dual Lefschetz map, which together with a counting operator give a representation of $U_q(\frak{sl}_2)$. There is some freedom in the choice of the adjoint, in fact a $q$-parameter, which corresponds to the $q$-parameter in $U_q(\frak{sl}_2)$. This works in quite some generality, but the motivating examples are again the (cominiscule) quantum flag manifolds endowed with their Heckenberger-Kolb calculus. This representation extends to calculi twisted by noncommutative vector bundles as presented here. In the quantum homogeneous space case the calculi admit a natural $L^2$-completion to a Hilbert space, to which the $U_q(\frak{sl}_2)$ representation can be extended to a representation by bounded operators. The representation of the (2,2)-SUSY algebra, as explained in Huybrechts, does not seem to directly $q$-deform, something more subtle happens which is still not well-understood.

An analogous formulation of noncommutative hypercomplex structures is most likely possible, but should probably wait until we actually have some examples :)

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