TITLE: Examples of representations of quantum groups
QUESTION [15 upvotes]: 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).

REPLY [3 votes]: 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 :)