Talk on finite-dimensional representations of quantum affine algebras

overview

• f-dim'l repns of affine Lie algebra
• f-dim'l repns of quantum affine algebra

$ \newcommand{\Z}{\mathbb Z} \newcommand{\C}{\mathbb C} \newcommand{\h}{\mathfrak{h}} \newcommand{\g}{\mathfrak{g}} \newcommand{\Lg}{L\g} \newcommand{\ghat}{\widehat{\g}} \newcommand{\uqg}{U_q(\g)} \newcommand{\uqghat}{U_q(\ghat)} \newcommand{\ev}{\operatorname{ev}} \newcommand{\eva}{\operatorname{ev}_{a}} \newcommand{\sltwo}{\operatorname{sl}_{2}} \newcommand{\la}{\lambda} \newcommand{\DP}{\mathbf{P}} $

simple Lie algebra

• $\g$ simple Lie algebra of rank $r$
• generators $\{e_i,f_i,h_i\}_{1\leq i \leq r}$
• classification of finite-dim'l irrep
• every finite-dim'l irrep is a highest weight repn
• bijection between finite-dim'l irrep and dominant integral weight

affine Lie algebra

• $\ghat\supset \g$, generators $\{e_i,f_i,h_i\}_{0\leq i \leq r}$
• every integrable irr. rep'n is a highest weight repn
• bijection between integrable irr. rep'n and dominant integral weight

loop algebra realization of affine Lie algebra

• loop algbera

$$\Lg=\g\otimes\mathbb{C}[t,t^{-1}]$$ $$[\alpha(m),\beta(n)]=[\alpha,\beta]\otimes t^{m+n},\qquad \alpha(m)=\alpha\otimes t^m$$

• affine Lie algebra is a central extension of loop algebra

$$ 0\to \mathbb{C}c \to \ghat \to \Lg \to 0 $$ with $$[\alpha(m),\beta(n)]=[\alpha,\beta]\otimes t^{m+n}+m\delta_{m,-n}\langle \alpha,\beta\rangle c$$ $$[c,x] =0, x\in \ghat$$

• $\langle \cdot,\cdot\rangle$ invariant form on $\g$

f-dim'l repns of $\Lg$

prop

Let $V$ be a f-dim'l repn of $\ghat$. Then $c$ acts as 0.

pf

abstract non-sense + highest weight theory for $\sltwo$. ■

• so we study f-dim'l repns of $\Lg=\ghat/\C c$

evaluation repn

• evaluation homomorphism $\eva : \Lg\to \g$ for $a\in \mathbb{C}^{\times}$ defined by

$$ \eva(x\otimes t^m)=a^m x\\ $$

• if $V$ is a $\g$-module, then the pull-back $V(a):=\eva^{*}(V)$ of $V$ is a $\Lg$-module.
  • i.e. for $(x\otimes f(t)).v = f(a)xv$

thm

1. Let $V_1,\dots, V_n$ be non-trivial irrep of $\g$. Then $V_1(a_1)\otimes \dots \otimes V_n(a_n)$ is an irrep of $\Lg$ iff $a_i\neq a_j $ for all $i\neq j$
2. Every irrep of $\Lg$ is of the form $V_1(a_1)\otimes \dots \otimes V_n(a_n)$
3. Let $V_1,\dots, V_n ,W_1,\dots, W_m$ are non-trivial irreps of $\g$. Then $V_1(a_1)\otimes \cdots \otimes V_n(a_n)$ and $W_1(b_1)\otimes \cdots \otimes W_m(b_m)$ are isomorphic if and only if $m = n$ and, up to a permutation of the indices, $V_i = Wi, a_i=b_i$ for every $i=1,\dots, n$

• Q. is an irrep of $\Lg$ a highest-weight representation?
• recall that there are two sets of generators of $\Lg$

1. as $\Lg=\ghat/\C c$, $\{e_i,f_i,h_i\}_{0\leq i \leq r}$ are generators of $\Lg$
2. loop generators $\{e_i\otimes t^s, f_i\otimes t^s, h_i\otimes t^s\}_{1\leq i \leq r, \qquad s\in \Z}$

• to define notion of highest weight representation, we need to choose "raising operators" that annihilate a highest weight vector
• hence there are two candidates for raising operators

1. $\{e_i\}_{0\leq i \leq r}$
2. $\{e_i\otimes t^s\}_{1\leq i \leq r, \, s\in \Z}$

example

• $\g=\sltwo$, $e_1,f_1,h_1$
• $\ghat$, $e_0,f_0,h_0,e_1,f_1,h_1$
• we can make identification

$$e_0=f_1\otimes t,\,f_0=e_1\otimes t^{-1},h_0=-h_1+c$$

• $V$ : 2-dim'l standard irrep of $\g$, basis $v_1, v_{-1}$,
• $v_1$ is h.w vector for $\g$; $e_1.v_1 = 0$
• fix $a\in \C^{\otimes}$
• how candidates for raising operators of $\Lg$ act on the evaluation representation $V(a)$?

1. $e_0.v_1 = a f_1.v_1 = a v_{-1}$
2. $(e_1\otimes t^{s}).v_1 = a^{s} (e_1.v_1) = 0$

• we can say $V(a)$ is a h.w representation of $\Lg$ with $\{e_1\otimes t^{s}\}_{s\in \Z}$ as raising operators
• in general, if we choose $\{e_i\otimes t^s\}_{1\leq i \leq r, \, s\in \Z}$ as raising operators, all $\Lg$-irreps become highest weight representation

how to write highest weights

• we already have a good parametrization for $\Lg$-irreps $\{(\la_1,a_1),\dots, (\la_n,a_n)\}$ with $\la_i$ dominant weights, $a_i\in \C^{\times}$ with $a_i\neq a_j$
• unsatisfactory since the concept of weights for $\g$ or $\ghat$ also has algebraic structure
• weight should encode eigenvalues of Cartan generators $\{h_{i,s}\}$
• we use a different Cartan generators
• define $P_{i,s}\in U(\h)$

$$ P_i(u):=\sum_{s=0}^{\infty}P_{i,s}u^s=\exp \left(-\sum_{s=1}^{\infty}\frac{h_{i,s}}{s}u^s\right) $$ with formal variable $u$

• it turns out

$$ P_{i,s}=\binom{h_i}{s}\otimes (-t)^s $$

• let $V$ be a $\g$-irrep of highest weight $\la$ with h.w. vector $v$. Then

$$ P_i(u).v=\sum_{s=0}^{\infty}(-a)^r\binom{\la(h_i)}{r}u^r v=(1-au)^{\la(h_i)}.v $$

• hence generating function of eigenvalues of $P_{i,s},\, s=0,1,\dots$ gives a polynomial
• we can say $V_1(a_1)\otimes \dots \otimes V_n(a_n)$ is a hightest weight representation whose highest weight is $r$-tuples of polynomials $(1-a_1u)^{\la_1(h_i)}\dots (1-a_nu)^{\la_n(h_i)}$ for each $i=1,\dots,r$
• we can now multiply weights

Drinfeld-Jimbo quantum groups

• let $q\in \C^{\times}$ not a root of 1
• $\g$ symmetrizable Kac-Moody algebra, i.e.
• $C=(c_{i,j})_{1\leq i,j\leq n}$ : generalized Cartan matrix
• there exists non-singular $D=\operatorname{diag}(s_1,\dots,s_n)$ s.t. $B=DC$ is symmetric
• fix unique coprime $s_1,\dots,s_n \in \Z_{>0}$, $q_i:=q^{s_i}$
• two important cases
  • $C$ of finite type (principal minors are $>0$)
  • $C$ of affine type (proper principal minors are $>0$, $\det C=0$)

• Drinfeld-Jimbo quantum group is $q$-deformation of $U(\g)$

def

$\uqg$ is the $\C$-algebra defined by generators $x_i^{+}, x_i^{-}, k_{i}^{\pm 1},\, (1\leq i\leq n)$ and relations ...

• $\uqg$ has a Hopf algebra structure

f-dim'l repns of quantum affine algebras

• $\g$ : simple Lie algebra of rank $r$ $I=\{1,\dots, r\}$
• quantum affine algebra $\uqghat$ = Drinfeld-Jimbo quantum group for affine Lie algebra $\ghat$

Drinfeld realization of $\uqghat$

Thm (Drinfeld-Beck)

$\uqghat$ has another presentation in terms of generators

• $x_{i,n}^{\pm}, i \in I, n \in \Z$
• $k_i^{\pm}, i \in I$
• $h_{i,n}^{\pm}, i \in I, n \neq 0$
• $C^{\pm 1}$

and relations ...

classification of irreps

• for each $i\in I$ let

$$ \begin{aligned} \phi_i^\pm(u) &= \sum_{n=0}^{\infty}\phi_{i,\pm n}^{\pm}u^{\pm n} \\ &= k^{\pm 1}_i \exp \left( \pm (q_i-q_i^{-1}) \sum_{n>0}h_{i,\pm n} u^{\pm n} \right) \end{aligned} $$

thm (Chari-Pressley)

Let $V$ be an irrep of $\uqghat$. Then $V$ is a $\ell$-highest weight (i.e. highest weight in the sense of Drinfeld realization). In other words, In other words, $V$ is generated by a vector $v$ such

• $x^+_{i,n}v=0$ for all $i \in I, n \in \Z$
• $C^{\pm 1}.v = v$
• $v$ is an eigenvector of $\phi_{i,n}^\pm$ for all $i \in I, n \in \Z$ and

\begin{equation} \phi_i^\pm(u)v = q_i^{\deg(P_i)}\frac{P_i(uq_i^{-1})}{P_i(uq_i)}v \end{equation} for some polynomial $P_i(u)\in \C[u]$, $P_i(0)=1$.

• There is a bijection between $r$-tuples polynomials and iso. class of irreps : $\DP=(P_i(u))_{i\in I}\mapsto V(\DP)$

example

• fundamental representation of $\uqghat$ corresponds to
  • $\DP=(1,1,\dots,1,1-au,1,\dots, 1),\, a\in \C^{\times}$, $i$-th position
• Kirillov-Reshetkhin modules correspond to
  • $\DP=(1,1,\dots,1,(1-au)(1-aq_i^2u)\dots (1-aq_i^{2(m-1)}u),1,\dots, 1),\, a\in \C^{\times}$, $i$-th position
  • hence parametrized by $(i,a,m)\in I\times \C^{\times} \times \Z_{\geq 0}$

KR-module

• type $A_1$
• consider the KR modules $W_{m,a}^{(1)}$
• the irreducible finite dimensional repn of $U_q(\ghat)$ that corresponds to the Drinfeld polynomial

$$ \mathcal{P}(u) =\prod _{i=1}^m \left(1-ua q^{m+1-2i}\right)\in \mathbb{C}[u] $$

• let $v_0^{(m)}$ be a highest weight vector and $\psi_{k}$ be the eigenvalue of $\phi_{1,k}$
• we've seen that

$$ \psi_{0} =q^m $$ and $$ \psi_{k} =a^k q^{k m} \left(q^m-q^{-m}\right),\,k>0 $$ so that $$ \psi_{1}(u)=\sum _{k=0}^{\infty } \psi_k u^k = q^{m}\frac{\mathcal{P}(u q^{-1})}{\mathcal{P}(u q)} $$

related items

• Talk on q-characters for beginners
• Affine Kac-Moody algebras as central extensions of loop algebras