Talk on finite-dimensional representations of quantum affine algebras
loop algebra realization of affine Lie algebra
- start with a semisimple Lie algebra $\mathfrak{g}$ with invariant form $\langle \cdot,\cdot\rangle $
- make a vector space from it
- construct the loop algbera
$$\hat{\mathfrak{g}}=\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]$$ $$\alpha(m)=\alpha\otimes t^m$$
- Add a central element to get a central extension and give a bracket
$$\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}c$$ $$[\alpha(m),\beta(n)]=[\alpha,\beta]\otimes t^{m+n}+m\delta_{m,-n}\langle \alpha,\beta\rangle c$$ $$[c,x] =0, x\in \hat{\mathfrak{g}}$$
classification
review of finite-dimensional representations of quantum affine algebras
Cartan generators
- $q_i = q^{d_i}$ for certain integers $d_i$
- for each $i\in I$ let
$$ \label{series} \begin{aligned} \phi_i^\pm(u) &= \sum_{n=0}^{\infty}\phi_{i,\pm n}^{\pm}u^{\pm n} \\ &= k^{\pm 1}_i \exp \left( \pm (q_i-q_i^{-1}) \sum_{n>0}h_{i,\pm n} u^{\pm n} \right) \end{aligned} $$
highest weight representation
- Every finite-dim'l irr rep'n of $U_q(\widehat{\mathfrak{g}})$ is a highest weight representation. In other words,
- it is generated by a highest weight vector $v$
- annihilated by $x^+_{i,n}, i \in I, n \in \Z$
- eigenvector of $\phi_{i,n}^\pm, i \in I, n \geq 0$
- Let $\phi_{i,n}^\pm v=\psi_{i,n}^\pm v$ for some $\psi_{i,n}^\pm\in \mathbb{C}$ and
$$ \psi_i^\pm(u) : = \sum_{n=0}^\infty \psi_{i,n}^\pm u^{\pm n}. $$
- thm [CP]
Let $V$ be a finite-dimensional irreducible representation of $U_q \widehat{\mathfrak g}$ of type 1 and highest weight $(\psi_{i,r}^{\pm})_{i\in I,r\in\Z}$. Then, there exists $\mathbf P=(P_i)_{i\in I}\in\mathcal P$ such that \begin{equation} \psi_i^\pm(u) = q_i^{\deg(P_i)}\frac{P_i(uq_i^{-1})}{P_i(uq_i)}. \end{equation} as an element of $\C[[u^{\pm 1}]]$.
Assigning to $V$ the set $\mathbf P$ defines a bijection between $\mathcal P$ and the set of isomorphism classes of finite-dimensional irreducible representations of $U_q \widehat{\mathfrak g}$ of type 1. The irreducible representation associated to $\mathbf P$ will be denoted by $V(\mathbf P)$.
- Note that in our notation the polynomial $P_i(u)$ corresponds to the polynomial $P_i(uq_i^{-1})$ in the notation of [CP]