2
$\begingroup$

Context

Let $V$ be a 2-dimensional evaluation representation of the quantum loop algebra $\mathcal{U}_{q}\left(\mathcal{L}\mathfrak{sl}_{2}\right)$ with $a=q$. Also, for $m\in\mathbb{Z}$, the Drinfeld generator $x_{m}^{+}$ ($x^{-}_{m}$) acts by $k^{m}X$ ($k^{m}Y$) on $V$.

We recall that $\mathcal{U}_{q}\left(\mathcal{L}\mathfrak{sl}_{2}\right)$ contains the Chevalley generators $X_{1}=x^{+}_{0}$, $X_{0}=k^{-1}x^{+}_{1}$, $Y_{1}=x^{-}_{0}$, $Y_{0}=x^{-}_{-1}k$ with the coproduct $$ \triangle\left(X_{0}\right)=X_{0}\otimes1+k^{-1}X_{0},\triangle\left(X_{1}\right)=X_{1}\otimes1+k\otimes X_{1},\triangle\left(k\right)=k\otimes k $$ $$ \triangle\left(Y_{0}\right)=Y_{0}\otimes k+1\otimes Y_{0},\triangle\left(Y_{1}\right)=Y_{1}\otimes k^{-1}+1\otimes Y_{1} $$ and for $m,m^\prime\ge0$ such that $m+m^\prime\ne0$: $$ \left[x_{-m}^{+},x_{-m^{\prime}}^{-}\right]=\phi_{-m-m^{\prime}}/\left(q^{-1}-q\right), $$ where $$ \phi^{-}\left(z\right)=k^{-1}\exp\left(\left(q^{-1}-q\right)\sum_{m>0}h_{-m}z^{-m}\right)=\sum_{m\ge0}\phi_{-m}z^{-m} $$

Problem

Consider $$ T\left(z\right)=\exp\left(\sum_{m>0}\frac{h_{-m}\left(q-q^{-1}\right)z^{-m}}{\left(q^{2m}-q^{-2m}\right)}\right) $$

Let $g\left(z\right)$ be the eigenvalue of $T\left(z\right)$ over $\left(V\right)_{q}$.

How to show that the eigenvalues of $\left(g\left(z\right)\right)^{-1}T\left(z\right)$ on $V\left[\left[z\right]\right]$ are $1$ and $1-zq^{-1}$?

Along with that, I would like to understand better how $T\left(z\right)$ acts on $V\left[\left[z\right]\right]$.

My thoughts so far

The $1$ eigenvalue is obvious, with $\left(V\right)_{q}$ being an eigenspace. I think that $\left(V\right)_{q}z$ is probably an eigenspace corresponding to eigenvalue $1-zq^{-1}$. However, I'm not sure how to get the result. Perhaps it can be done using Drinfeld polynomials or considering eigenvalues of $h_{-m}$ and noticing something about $T\left(z\right)$.

$\endgroup$
3
$\begingroup$

For the meaning of $T(z)$, see the paper of Frenkel and Hernandez (http://arxiv.org/abs/1308.3444, version 4) Example 5.6. The eigenvalues of $T(z)$ are related to the so-called Frenkel-Reshetikhin q-character of $V$. See Proposition 5.8 of that paper for the precise statement and the proof.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.