Quantized universal enveloping algebra

수학노트
이동: 둘러보기, 검색

개요

  • \(U_{q}(\mathfrak{g})\) : Kac-Moody 대수의 UEA \(U(\mathfrak{g})\) 의 deformation
  • 호프 대수(Hopf algebra)의 구조를 가짐
  • 양자군 (quantum group)의 예


Cartan datum

  • Cartan datum \((A,P^{\vee},P,\Pi^{\vee},\Pi)\)
  • \(A=(a_{ij})_{i,j\in I}\) symmetrizable GCM
    • \(D=\operatorname{diag}(s_i\in\mathbb{Z}_{\geq 0})_{i \in I}\) diagonal matrix s.t. DA is symmetric
  • \(P^{\vee}=(\bigoplus_{i\in I}\mathbb{Z}h_{i})\bigoplus(\bigoplus_{j=1}^{\operatorname{corank}(A)}\mathbb{Z}d_{j})\) : dual weight lattice
  • \(\mathfrak{h}=\mathbb{Q}\otimes_{\mathbb{Z}} P^{\vee}\) : Cartan subalgebra
  • \(P=\{\lambda\in\mathfrak{h}^{*}|\lambda(P^{\vee})\subset \mathbb{Z}\}\) : weight lattice
  • \(\Pi^{\vee}=\{h_{i}|i\in I\}\) : simple coroots
  • \(\Pi=\{\alpha_{i}\in\mathfrak{h}^{*}|i\in I, \alpha_{i}(h_j)=a_ {ji}\}\) : simple roots
  • \((\cdot|\cdot)\) symmetric bilinear form on \(\mathfrak{g}^{*}\)
  • \(s_{i}=\frac{(\alpha_{i}|\alpha_{i})}{2}\in \mathbb{Z}_{>0}\)
  • q: indeterminate
  • \(q_i=q^{s_{i}}\)
  • q-초기하급수(q-hypergeometric series)와 양자미적분학(q-calculus)


정수의 q-analogue

  • 정수 n에 대하여 다음과 같이 정의\[[n]_{q_i} =\frac{q_ {i}^{n}-q_ {i}^{-n}}{q_i-q_i^{-1}} \]\[[0]_{q_i} =1\]\[[n]_{q_i}!=[n]_ {q_i}[n]_ {q_i}\cdots [n]_{q_i}\]\[{m \choose n}_{q_{i}}=\frac{[m]_{q!}}{[n]_{q_{i}!}[m-n]_{q_i}!}\]
  • 극한 \(q \to 1\)


quantized universal enveloping algebra \(U_{q}(\mathfrak{g})\)

  • 생성원 \(e_i,f_i , (i\in I)\), \(q^{h} (h\in P^{\vee})\)
  • 관계식
    • \(q^0=1\)
    • \(q^{h}q^{h'}=q^{h+h'}\)
    • \(e_if _j-f_je _i=\delta_{i,j}\frac{k_i-k_i^{-1}}{q_i-q_i^{-1}}\) 여기서 \(k_{i}=q^{h_is _i}\)
    • \(q^he_jq^{-h}=q^{\alpha_j(h)}e_j\)
    • \(q^hf_jq^{-h}=q^{-\alpha_j(h)}f_j\)
    • \(\sum_{k=0}^{1-a_{i,j}}(-1)^k \binom{1-a_{i,j}}{k}_{q_{i}}e_ {i}^{1-a_{i,j}-k}e_{j}e_ {i}^k=0\) (\(i \neq j\))
    • \(\sum_{k=0}^{1-a_{i,j}}(-1)^k \binom{1-a_{i,j}}{k}_{q_{i}}f_ {i}^{1-a_{i,j}-k}f_{j}f_ {i}^k=0\) (\(i \neq j\))



호프 대수 구조

  • comultiplication

\[\Delta : U_{q}(\mathfrak{g}) \to U_{q}(\mathfrak{g}) \otimes U_{q}(\mathfrak{g})\] \[\Delta(q^{h}) =q^{h}\otimes q^{h}\] \[\Delta(e_i)=e_i\otimes k_i+1\otimes e_i\] \[\Delta(f_i)=f_i\otimes 1+ k_i^{-1}\otimes f_i\]

  • counit

\[\epsilon(q^{h}) =1\] \[\epsilon(e_i)=\epsilon(f_i)=0\]

  • antipode

\[S(q^h) = q^{-h}\] for \(x \in \mathfrak{g}\) \[S(e_i) =-e_ik_i^{-1}, S(f_i)=-k_if_i\]


극한 $q \to 1$


역사



메모



관련된 항목들

 


사전 형태의 자료