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I am trying to solve the following problem, which is related to relatively recent results, but I am not sure how to do it.

Problem

In this problem, $\mathfrak{g}=\mathfrak{sl}_{2}$. We study finite-dimensional representations of loop algebra $L\mathfrak{g}$. Let $a\in\mathbb{C}^{*}$ and $\tilde{\lambda}_{a}\in\left(L\mathfrak{h}\right)^{*}$ defined by $\tilde{\lambda}_{a}\left(t^{r}H\right)=a^{r}$ for all $r\in\mathbb{Z}$.

  • Show that the representation $L\left(\tilde{\lambda}_{a}\right)$ of the $l$-highest weight is finite-dimensional [one can use the morphism of evaluation $ev_{a}:t^{t}\otimes x\rightarrow a^{r}x$]. Specify its dimension and its $l$-character.
  • Show that the module $P_{a}=L\left(\tilde{\lambda}_{a}\right)\otimes L\left(\tilde{\lambda}_{a}\right)$ is not simple.
  • Is $P_{a}$ semi-simple?

Some results of my attempts to solve it

I am not sure what are the definitions of $l$-weights and $l$-characters of $L\mathfrak{g}$, but I intuitively applied the formulas $\left[t^{r}\otimes x,t^{k}\otimes y\right]=t^{r+k}\left[x,y\right]$ and $\left[E^{\pm},H\right]=\pm2H$ to the assumed expression for highest $l$-weight and highest $l$-weight vector $\left(t^{r}\otimes H\right)\left(\Lambda\right)=a^{r}\Lambda$.

From the calculations, it appears that the "weights" are $a^{r}$, $a^{r}-2t^{r}$, $a^{r}-4t^{r}$, $\cdots$, but this does not look right, because I do not expect $t$ to be part of the expressions for $l$-weights (perhaps if an expression for $l$-weight contains $t$, it is not even a weight, but I have never dealt with $l$-weights before, so maybe I'm wrong).

References

[1] Chari, Vyjayanthi, Integrable representations of affine Lie-algebras, Invent. Math. 85 (1986), no. 2,317–335.

[2] Chari, Vyjayanthi and Hernandez, David, Beyond Kirillov-Reshetikhin modules, 49–81, Contemp. Math., 506, Amer. Math. Soc., Providence, RI, 2010.

[3] Frenkel, Edward, Reshetikhin, Nicolai, The q-characters of representations of quantum affine algebras and deformations of W-algebras, Contemp. Math., 248, Amer. Math. Soc., Providence, RI, 1999.

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    $\begingroup$ if we know $m_1, m_2 \in \mathbb{Z}[Y_{i,a}]_{i \in I, a \in \mathbb{C}^*}$ explicitly, then it is easy to check that $L(m_1) \otimes L(m_2)$ is simple or not by using the Frenkel-Mukhin algorithm. In the case of $\mathfrak{sl}_2$ it is easy to check that $L(m_1) \otimes L(m_2)$ is simple or not for general $m_1, m_2$. $\endgroup$ Apr 12, 2015 at 9:04

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