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$\mathfrak{sl}(2)$ (over $\mathbb{C}$) with basis $E_\pm, H$ with commutation relations $$ [H,E_{\pm}]=\pm 2 E_\pm,\quad [E_+,E_-]=H $$ admits the well-known representation on $\mathbb{C}[x]$ with $$ E_+ = \partial_x,\quad E_- = -x^2 \partial_x + s\,x,\quad H = -x \partial_x - s $$ where $\partial_x = \frac{d}{dx}$. This representation is highest-weight with highest-weight vector given by $1\in \mathbb{C}[x]$. The parameter $s$ is free to take any value in $\mathbb{C}$. Furthermore this differential operator realisation can also be used if one changes the space of functions on which we act. For example, the principle series representations can be realised in this way.

Question: Can highest-weight representations of $U_q(\mathfrak{sl}(2))$ be realized in this way? I am aware that $U_q(\mathfrak{sl}(2))$ admits a representation on the quantum plane $\mathbb{C}_q[x,y]$ - this is not the representation I am looking for since that has a direct counterpart for usual $\mathfrak{sl}(2)$, which is not the representation I described at the beginning.

I am familiar with the textbooks by Chari and Pressley and by Kassel but have not come across such a representation.

I have tried to construct it directly, granted using a rather naive approach and substituting the differential operator $\partial_x$ with the $q$-differential operator $D_q$. This did not work due to the fact that the Liebniz rule for q-differentiation involves a multiplication by $q$ in the functions which are not differentiated. q-derivative

Does such a representation exist? If so please provide a reference - I am also interested in the generalisation to higher-rank cases.

Edit: The representation on $\mathbb{C}_q[x,y]$ is not the representation I am looking for as in the $q\rightarrow 1$ limit this reduces to a representation on $\mathbb{C}[x,y]$, not $\mathbb{C}[x]$ as I described in the initial paragraph of the question. The representation I am looking for should reduce to the one described initially for $q\rightarrow 1$.

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  • $\begingroup$ I am little confused on the follwoing point of your question: you say that "I am aware that $U_q(\mathfrak{sl}(2))$ admits a representation on the quantum plane $\mathbb{C}_q[x,y]$ - this is not the representation I am looking for since that has a direct counterpart for usual $\mathfrak{sl}(2)$". So, are you looking for representations which have no direct counterparts on the $sl_q(2)$ representations? $\endgroup$
    – Konstantinos Kanakoglou
    Commented Jun 14, 2021 at 20:23
  • $\begingroup$ In other words, you want a differential realization of $sl_q(2)$, but one not limitting to the differential realization of the $sl(2)$ given in the OP ? $\endgroup$
    – Konstantinos Kanakoglou
    Commented Jun 14, 2021 at 20:24
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    $\begingroup$ @KonstantinosKanakoglou Thank you for the comments - I have added clarifying details to the initial post. $\endgroup$
    – Yamero
    Commented Jun 15, 2021 at 8:06

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Although i have some doubts as to what the OP is exactly looking for (see my comments above), i hope that the following will be of some interest for its purposes. In:

the author introduces a unified scheme for constructing differential operator realizations for any irreducible representation of $U_q(sl(n))$. These come as $q$-deformations of the corresponding realizations for $sl(n)$ irreducible representations.
What is actually done in this paper, is the generalization of results shown in:

where formally similar realizations are obtained as $q$-deformations of the differential/difference realizations of totally symmetric $sl(2)$ and $sl(3)$ representations. (in my understanding, though i admit i have given just a quick look at this last reference, the realizations given there are some quotient of the $U_q(sl(2))$ representations on the quantum plane $\mathbb{C}_q[x,y]$ mentioned in the OP).

Edit: One can find similar deformed differential/difference realizations for $q$-deformations of other Lie algebras as well. For some more cases in a similar spirit one can see for example:

Edit 2: For the completeness of this answer, i think it might be interesting to mention that there are more general, systematic ways of building $q$-deformed differential realizations of the various deformations $U_q$ of the UEA of (super)Lie algebras. These are using some $q$-deformed CCR/CAR (super)algebra $W_q$ (that is $q$-deformed Weyl/Clifford (super)algebras) in order to realize the quantum group with an algebra $D_q$ of deformed differential operators, acting on spaces spanned by polynomials mixing usual (commutative) and Grassman (non-commutative) variables, according to the general scheme: $$ U_q\stackrel{(I)}{\longrightarrow} W_q\stackrel{(II)}{\longrightarrow} D_q $$ In this sense, the $D_q$ representations are pulled back to representations of the quantum group $U_q$.
There is a significant amount of literature on similar techniques (usually found in mathematical physics journals).
For example, one can find realizations (of type $(I)$) of quantum groups in terms of $q$-deformed bosons (or fermions) at:

And deformed differential realizations (of type $(II)$) of $q$-bosons at:

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    $\begingroup$ Thank you, the representation described in the first paper you linked is precisely what I was looking for! $\endgroup$
    – Yamero
    Commented Jun 15, 2021 at 8:07

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