Talk on finite-dimensional representations of quantum affine algebras

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Pythagoras0 (토론 | 기여)님의 2020년 11월 13일 (금) 23:14 판
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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 alg assoc. to \(A\) is the Lie alg 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 \(C\) = f-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 \(C\) = aff Lie alg
      • 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 alg

  • \(\g\) simple Lie alg of rank \(r\)
  • generators \(\{e_i,f_i,h_i\}_{1\leq i \leq r}\)
  • classification of f-dim'l irrep
  • every f-dim'l irrep \(V\) is a h.w. rep
    • \(\exists v\in V\) such that \(V\) is generated by \(v\) and \(e_i.v=0\)
  • bijection between f-dim'l irrep and dominant integral wt
    • i.e. \(r\)-tuple of non-negative integers \((n_1,\dots, n_r)\) given by \(h_iv=n_iv\)


aff Lie alg

  • \(\ghatp\supset \g\), 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 wts and they are h.w. reps and infinite dim'l except for trivial one (integrable means \(e_i,f_i\) are locally nilpotent for all \(i\))
  2. \(\ghatp\) has non-trivial f-diml reps and they cannot be extended to reps of \(\ghat\)
  3. affinization of f-dim'l rep of \(\ghatp\) -> integrable rep of \(\ghat\) not in \(\O\)

loop alg realization of aff Lie alg

  • loop algbera

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

  • aff Lie alg is a central extension of loop alg

\[ 0\to \mathbb{C}c \to \ghatp \to \Lg \to 0 \] 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 + h.w. 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

monoid of highest weights

  • we already have a good parametrization for \(\Lg\)-irreps \(\{(\la_1,a_1),\dots, (\la_n,a_n)\}\) with \(\la_i\) dominant wts, \(a_i\in \C^{\times}\) with \(a_i\neq a_j\)
  • unsatisfactory since the concept of wts for \(\g\) or \(\ghat\) also has algebraic structure (addition)
  • wt 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 h.w. \(\la\) 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 wts (wts form a monoid)


Drinfeld-Jimbo quantum groups

  • \(\g\) symmetrizable Kac-Moody alg 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 alg \(U(\g)\) of \(\g\)
  • \(\uqg\) has a Hopf alg structure

f-dim'l reps of quantum aff alg

  • quantum aff alg = Drinfeld-Jimbo quantum group for aff GCM \(C\)
  • assume \(C\) is untwisted aff type whose underlying finite type has rank \(r\)
  • Let \(I=\{1,\dots, r\}\)
  • \(\uqghat\) is the \(\C\)-alg 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

def

A \(\uqghatp\)-module \(M\) is integrable (also called type 1) if (1) as \(\uqg\)-module, \(M\) has decomposition \(M=\oplus_{\la}M_{\la}\) where \(M_{\la}=\{u\in M: k_i u = q_{i}^{\la(h_i)}u, \, i\in I\}\), and (2) \(e_i,f_i\) acts locally nilpotently for all \(i=0,1,\dots, r\).


  • 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 f-dim'l int. irrep of \(\uqghatp\). Then \(V\) is a \(\ell\)-highest wt (i.e. h.w. 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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