Quantized universal enveloping algebra
둘러보기로 가기
검색하러 가기
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.
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 \(q \to 1\)
- http://mathoverflow.net/questions/92046/quantum-group-uqsl2
- \([a,b]=\lambda b\) 이면, \(q^a b q^{-a}=q^{\lambda} b\)
- 베이커-캠벨-하우스도르프 공식 참조
역사
메모
- http://mathoverflow.net/questions/5538/why-drinfeld-jimbo-type-quantum-groups
- Math Overflow http://mathoverflow.net/search?q=
관련된 항목들
사전 형태의 자료
- http://ko.wikipedia.org/wiki/
- http://en.wikipedia.org/wiki/Quantum_group
- http://en.wikipedia.org/wiki/Quantum_affine_algebra
메타데이터
위키데이터
- ID : Q2122223
Spacy 패턴 목록
- [{'LOWER': 'quantum'}, {'LEMMA': 'group'}]