"호프 대수(Hopf algebra)"의 두 판 사이의 차이
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(피타고라스님이 이 페이지의 위치를 <a href="/pages/12529252">양자군 (quantum group)</a>페이지로 이동하였습니다.) |
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| 53번째 줄: | 53번째 줄: | ||
** comultiplication and counit are a ring maps | ** comultiplication and counit are a ring maps | ||
** multiplication and unit are a coring maps | ** multiplication and unit are a coring maps | ||
| − | ** antipode <math>\mu \circ (\operatorname{id}\otimes S) \circ \Delta = \mu \circ (S\otimes \operatorname{id}) \circ \Delta=1 \circ \epsilon</math> | + | ** antipode <math>\mu \circ (\operatorname{id}\otimes S) \circ \Delta = \mu \circ (S\otimes \operatorname{id}) \circ \Delta=1 \circ \epsilon</math>, 많은 경우는 antihomomorphism 즉, <math>S(ab)=S(b)S(a)</math> |
| 67번째 줄: | 67번째 줄: | ||
* one wants to have the following H-modules <math>V\otimes W</math> and <math>V^{*}</math> | * one wants to have the following H-modules <math>V\otimes W</math> and <math>V^{*}</math> | ||
* For Hopf algebra, we can construct them as H-modules | * For Hopf algebra, we can construct them as H-modules | ||
| + | * counit - trivial representations | ||
* tensor product<br><math>a.(v\otimes w)= \Delta(a)(v\otimes w)</math><br> | * tensor product<br><math>a.(v\otimes w)= \Delta(a)(v\otimes w)</math><br> | ||
* dual representation<br> For <math>f\in V^{*}</math>, <math>(a.f)(v)= f(S(a).v)</math><br> | * dual representation<br> For <math>f\in V^{*}</math>, <math>(a.f)(v)= f(S(a).v)</math><br> | ||
| − | + | * the category of representations has a monoidal structure with duals | |
| − | |||
| 78번째 줄: | 78번째 줄: | ||
<h5>예 : group ring</h5> | <h5>예 : group ring</h5> | ||
| − | <math>H=\mathbb{F}G</math> : group algebra of G over F | + | * <math>H=\mathbb{F}G</math> : group algebra of G over F |
| − | |||
* multiplication and identity element<br><math>m : \mathbb Z[G]\otimes \mathbb Z[G] \to \mathbb Z[G]</math><br> | * multiplication and identity element<br><math>m : \mathbb Z[G]\otimes \mathbb Z[G] \to \mathbb Z[G]</math><br> | ||
* comultiplication<br><math>\Delta : \mathbb Z[G] \to \mathbb Z[G]\otimes \mathbb Z[G] </math><br><math>g \mapsto g\otimes g</math><br> | * comultiplication<br><math>\Delta : \mathbb Z[G] \to \mathbb Z[G]\otimes \mathbb Z[G] </math><br><math>g \mapsto g\otimes g</math><br> | ||
| 91번째 줄: | 90번째 줄: | ||
<h5>예 : UEA</h5> | <h5>예 : UEA</h5> | ||
| − | * simple Lie algebra g | + | * simple Lie algebra <math>\mathfrak{g}</math> |
* <math>U(\mathfrak{g})</math> | * <math>U(\mathfrak{g})</math> | ||
* comultiplication (this explains why the tensor product of <math>U(\mathfrak{g})</math>-modules is defined as known)<br><math>\Delta : U(\mathfrak{g}) \to U(\mathfrak{g})\otimes U(\mathfrak{g}) </math><br><math>\Delta(x) =x\otimes 1+ 1 \otimes x</math> for <math>x \in \mathfrak{g}</math><br><math>\Delta(1)=1\otimes 1</math><br> | * comultiplication (this explains why the tensor product of <math>U(\mathfrak{g})</math>-modules is defined as known)<br><math>\Delta : U(\mathfrak{g}) \to U(\mathfrak{g})\otimes U(\mathfrak{g}) </math><br><math>\Delta(x) =x\otimes 1+ 1 \otimes x</math> for <math>x \in \mathfrak{g}</math><br><math>\Delta(1)=1\otimes 1</math><br> | ||
| 122번째 줄: | 121번째 줄: | ||
<h5>메모</h5> | <h5>메모</h5> | ||
| − | + | * [http://sbseminar.wordpress.com/2007/10/07/group-hopf-algebra/ Group = Hopf algebra] , Scott Carnahan, Secret Blogging Seminar, October 7, 2007 | |
| + | * http://mathoverflow.net/questions/101739/is-there-any-progress-on-the-theory-in-the-paper-geometric-methods-in-representa<br> | ||
* Math Overflow http://mathoverflow.net/search?q= | * Math Overflow http://mathoverflow.net/search?q= | ||
2012년 8월 26일 (일) 06:22 판
이 항목의 수학노트 원문주소
개요
- 호프 대수(Hopf algebra) = bi-algebra with an antipoe
- '군(group)' (군론(group theory) 항목 참조) 개념의 일반화
- 양자군의 이론에서 중요한 역할
- 양자군(quantum group) = non co-commutative quasi-triangular Hopf algebra
군(group) 의 정의 : abstract nonsense
- 군의 정의를 abstract nonsense를 사용하여 표현하기
- a group is a set \(G&bg=ffffff&fg=000000&s=0\) equipped with
- a multiplication map \(\mu: G \otimes G \to G\)
- an inversion map \(S: G \to G\)
- an identity element \(1:+*+\to+G&bg=ffffff&fg=000000&s=0\), where \(*&bg=ffffff&fg=000000&s=0\) is a one point set
- \(\epsilon:+G+\to+*&bg=ffffff&fg=000000&s=0\) (trivial representation, counit)
- \(\Delta: G \to G \otimes G\), diagonal map: \(g \mapsto g\otimes g\)
- 일반적으로 군을 정의할 때 드러나지 않는 \(\epsilon:+G+\to+*&bg=ffffff&fg=000000&s=0\) , \(\Delta:+G+\to+G+\times+G&bg=ffffff&fg=000000&s=0\)를 도입함으로써, 군을 abstract nonsense 만으로 표현할 수 있게 된다
- 결합법칙
\(\mu \circ (\mu \otimes \operatorname{id})=\mu \circ (\operatorname{id}\otimes \mu)\) - 역원에 대한 조건 (원소에 그 역원을 곱하면 항등원을 얻는다)
\(\mu \circ (\operatorname{id}\otimes S) \circ \Delta = \mu \circ (S\otimes \operatorname{id}) \circ \Delta=1 \circ \epsilon\), i.e., multiplying an element with its inverse yields the unit.
호프 대수(Hopf algebra) 의 정의
- Hopf algebra = an algebra with unit and three maps = bialgebra with an antipode
- Given a commutative ring \(R&bg=ffffff&fg=000000&s=0\), a Hopf algebra over \(R&bg=ffffff&fg=000000&s=0\) is a six-tuple \((G, \mu, 1, S, \epsilon, \Delta)\),
- \(G&bg=ffffff&fg=000000&s=0\)is an \(R&bg=ffffff&fg=000000&s=0\)-module
- \(\mu: G \otimes_R G \to G\) is a multiplication map
- \(1:+R+\to+G&bg=ffffff&fg=000000&s=0\) is a unit
- \(S: G \to G\) is called the antipode
- \(\epsilon:+G+\to+R&bg=ffffff&fg=000000&s=0\) is a counit
- \(\Delta:+G+\to+G+\otimes_R+G&bg=ffffff&fg=000000&s=0\) is called comultiplication.
- These are required to satisfy relations
- \((G,\mu,1)\) ring
- \((G,\Delta,\epsilon)\) coring (just turn all the previous arrows around)
- comultiplication and counit are a ring maps
- multiplication and unit are a coring maps
- antipode \(\mu \circ (\operatorname{id}\otimes S) \circ \Delta = \mu \circ (S\otimes \operatorname{id}) \circ \Delta=1 \circ \epsilon\), 많은 경우는 antihomomorphism 즉, \(S(ab)=S(b)S(a)\)
표현론에서 유용한 점
- H : Hopf algebra
- V,W : H-modules
- one wants to have the following H-modules \(V\otimes W\) and \(V^{*}\)
- For Hopf algebra, we can construct them as H-modules
- counit - trivial representations
- tensor product
\(a.(v\otimes w)= \Delta(a)(v\otimes w)\) - dual representation
For \(f\in V^{*}\), \((a.f)(v)= f(S(a).v)\) - the category of representations has a monoidal structure with duals
예 : group ring
- \(H=\mathbb{F}G\) : group algebra of G over F
- multiplication and identity element
\(m : \mathbb Z[G]\otimes \mathbb Z[G] \to \mathbb Z[G]\) - comultiplication
\(\Delta : \mathbb Z[G] \to \mathbb Z[G]\otimes \mathbb Z[G] \)
\(g \mapsto g\otimes g\) - counit
\(\epsilon(g)=1\) - antipode
\(S(g)=g^{-1}\)
예 : UEA
- simple Lie algebra \(\mathfrak{g}\)
- \(U(\mathfrak{g})\)
- comultiplication (this explains why the tensor product of \(U(\mathfrak{g})\)-modules is defined as known)
\(\Delta : U(\mathfrak{g}) \to U(\mathfrak{g})\otimes U(\mathfrak{g}) \)
\(\Delta(x) =x\otimes 1+ 1 \otimes x\) for \(x \in \mathfrak{g}\)
\(\Delta(1)=1\otimes 1\)
- counit
\(\epsilon(x) =0\) for \(x \in \mathfrak{g}\)
\(\epsilon(1) =1\) - antipode
\(S(x) = -x\) for \(x \in \mathfrak{g}\)
\(S(1) =1\) - quantized universal enveloping algebra
역사
메모
- Group = Hopf algebra , Scott Carnahan, Secret Blogging Seminar, October 7, 2007
- http://mathoverflow.net/questions/101739/is-there-any-progress-on-the-theory-in-the-paper-geometric-methods-in-representa
- Math Overflow http://mathoverflow.net/search?q=
관련된 항목들
수학용어번역
- 단어사전
- 발음사전 http://www.forvo.com/search/
- 대한수학회 수학 학술 용어집
- 한국통계학회 통계학 용어 온라인 대조표
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- 남·북한수학용어비교
- 대한수학회 수학용어한글화 게시판
매스매티카 파일 및 계산 리소스
- http://www.wolframalpha.com/input/?i=
- http://functions.wolfram.com/
- NIST Digital Library of Mathematical Functions
- Abramowitz and Stegun Handbook of mathematical functions
- The On-Line Encyclopedia of Integer Sequences
- Numbers, constants and computation
- 매스매티카 파일 목록
사전 형태의 자료
- http://ko.wikipedia.org/wiki/호프_대수
- [1]http://en.wikipedia.org/wiki/Hopf_algebra
- http://en.wikipedia.org/wiki/Coalgebra
- Encyclopaedia of Mathematics
- NIST Digital Library of Mathematical Functions
- The World of Mathematical Equations
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