Talk on finite-dimensional representations of quantum affine algebras

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imported>Pythagoras0님의 2019년 7월 14일 (일) 17:31 판 (→‎f-dim'l reps of quantum affine algebras)
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overview

  • f-dim'l reps of affine Lie algebra
  • f-dim'l reps 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{\ghatp}{\ghat'} \newcommand{\uqg}{U_q(\g)} \newcommand{\uqghat}{U_q(\ghat)} \newcommand{\uqghatp}{U_q'(\ghat)} \newcommand{\ev}{\operatorname{ev}} \newcommand{\eva}{\operatorname{ev}_{a}} \newcommand{\sltwo}{\operatorname{sl}_{2}} \newcommand{\la}{\lambda} \newcommand{\DP}{\mathbf{P}} \newcommand{\O}{\mathcal{O}} $


Kac-Moody algebra

  • A generalized Cartan matrix is a square matrix $A = (a_{ij})$ with integer entries such that
  1. For diagonal entries, $a_{ii} = 2$.
  2. For non-diagonal entries, $a_{ij} \leq 0$.
  3. $a_{ij} = 0$ if and only if $a_{ji} = 0$
  • A GCM $C$ is called indecomposable if the graph with vertex set $I$ and edge set $\{(i,j): a_{ij}<0\}$ is connected
  • Kac-Moody algebra assoc. to $A$ is the Lie algebra generated by \(e_i,f_i ,h_i(i\in I)\), \(d_j,(j\in\{1,\dots, \operatorname{corank}(A)\}\) with relations
  • two important indec. GCMs
    • $C$ of finite type (principal minors are $>0$); KM of $C$ = fin-dim'l simple Lie alg
      • $A_l,B_l,C_l,D_l,E_6,E_7,E_8,F_4,G_2$
    • $C$ of affine type (proper principal minors are $>0$, $\det C=0$); KM of $C$ = affine Lie algebra
      • untwisted : $A_l^{(1)},B_l^{(1)},C_l^{(1)},D_l^{(1)},E_6^{(1)},E_7^{(1)},E_8^{(1)},F_4^{(1)},G_2^{(1)}$
      • twisted : $A_2^{(2)},A_{2l}^{(2)},A_{2l-1}^{(2)}, D_{l+1}^{(2)}, E_{6}^{(2)}, D_{4}^{(3)}$
  • today, 'affine Lie alg' means 'untwisted affine Lie alg'

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 $V$ is a highest weight rep
    • $\exists v\in V$ such that $V$ is generated by $v$ and $e_i.v=0$
  • bijection between finite-dim'l irrep and dominant integral weight
    • i.e. $r$-tuple of non-negative integers $(n_1,\dots, n_r)$ given by $h_iv=n_iv$


affine Lie algebra

  • $\ghatp\supset \g$, generators $\{e_i,f_i,h_i\}_{0\leq i \leq r}$
  • $\ghat\supset \g$, generators $\{e_i,f_i,h_i\}_{0\leq i \leq r}$ and $d$
  1. integrable irreps of $\ghat$ in a category $\O$ are in bijection with dominant integral weights and they are h.w. rep and infinite dim'l except for trivial one
  2. $\ghatp$ has non-trivial fin-diml reps and they cannot be extended to reps of $\ghat$
  3. affinization of fin-dim'l rep of $\ghatp$ -> integrable rep of $\ghat$ not in $\O$


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

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

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


f-dim'l reps of $\Lg$

prop

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

pf

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

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


evaluation rep

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

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

  • 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 h.w. rep?
  • recall that there are two sets of generators of $\Lg$
  1. as $\Lg=\ghatp/\C c$, $\{e_i,f_i,h_i\}_{0\leq i \leq r}$ 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 h.w. rep, need to choose "raising operators" that annihilate a h.w. 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

  • $\g=\sltwo$, $e_1,f_1,h_1$
  • $\ghatp$, $e_0,f_0,h_0,e_1,f_1,h_1$
  • 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 $a\in \C^{\times}$
  • how candidates for raising operators of $\Lg$ act on the evaluation rep $V(a)$?
  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 rep 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 h.w. rep

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 $P_{i,s}^{\pm}\in U(\h)$

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

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

  • 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 h.w. rep whose h.w. is $r$-tuple of polys $(1-a_1u)^{\la_1(h_i)}\dots (1-a_nu)^{\la_n(h_i)}$ for each $i=1,\dots,r$
  • we can now multiply weights (weights form a monoid)


Drinfeld-Jimbo quantum groups

  • $\g$ symmetrizable Kac-Moody algebra with GCM $C$
  • there exists non-singular $D=\operatorname{diag}(d_i)_{i\in I}$ s.t. $B=DA$ is symmetric
  • fix unique coprime $s_1,\dots,s_n \in \Z_{>0}$, $q_i:=q^{d_i}$
  • let $q\in \C^{\times}$ not a root of 1
  • Drinfeld-Jimbo quantum group $\uqg$ is $q$-deformation of the universal enveloping algebra $U(\g)$ of $\g$
  • $\uqg$ has a Hopf algebra structure

f-dim'l reps of quantum affine algebras

  • quantum affine algebra = Drinfeld-Jimbo quantum group for affine GCM $C$
  • assume $C$ is untwisted affine type whose underlying finite type has rank $r$
  • Let $I=\{1,\dots, r\}$
  • $\uqghat$ is the $\C$-algebra defined by generators $x_i^{+}, x_i^{-}, q^{h_i}\, (i\in \{0,1,\dots,r\})$ and $q^{d}$ and relations
  • $\uqghatp$ is subalg of $\uqghat$ generated by $x_i^{+}, x_i^{-}, k_i:=q_i^{h_i}\, (i\in \{0,1,\dots,r\})$

Drinfeld realization of $\uqghat$

Thm (Drinfeld-Beck)

Then $\uqghatp$ 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 $i\in I, n\in \Z$, define $\phi^{\pm}_{i,n}$ by

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

thm (Chari-Pressley)

Let $V$ be an irrep of $\uqghatp$. Then $V$ is a $\ell$-highest weight (i.e. highest weight in the sense of Drinfeld realization). 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 irreps : $\DP=(P_i(u))_{i\in I}\mapsto V(\DP)$

example

  • fundamental rep $V_{i,a}$ of $\uqghatp$ corresponds to
    • $\DP=(1,1,\dots,1,1-au,1,\dots, 1),\, a\in \C^{\times}$, $i$-th position
  • Kirillov-Reshetkhin module $W^{(i)}_{a,m}$ with $(i,a,m)\in I\times \C^{\times} \times \Z_{\geq 0}$ corresponds 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

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