"호프 대수(Hopf algebra)"의 두 판 사이의 차이

2021년 2월 17일 (수) 03:50 기준 최신판

개요

호프 대수(Hopf algebra) = bi-algebra with an antipode
'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 \(G\) equipped with
- a multiplication map \(\mu: G \otimes G \to G\)
- an inversion map \(S: G \to G\)
- an identity element \(1:*\to G\), where \(*\) is a one point set
- \(\epsilon:G\to *\) (trivial representation, counit)
- \(\Delta: G \to G \otimes G\), diagonal map \(g \mapsto g\otimes g\)
일반적으로 군을 정의할 때 드러나지 않는 \(\epsilon:G \to *\) , \(\Delta:G \to G \times G\)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)\[\mu \circ (\operatorname{id}\otimes S) \circ \Delta = \mu \circ (S\otimes \operatorname{id}) \circ \Delta=1 \circ \epsilon\]

호프 대수(Hopf algebra) 의 정의

Hopf algebra = an algebra with unit and three maps = bialgebra with an antipode
Given a commutative ring \(R\), a Hopf algebra over \(R\) is a six-tuple \((G, \mu, 1, S, \epsilon, \Delta)\),
- \(G\)is an \(R\)-module
- \(\mu: G \otimes_R G \to G\) is a multiplication map
- \(1:R \to G\) is a unit
- \(S: G \to G\) is called the antipode
- \(\epsilon: G \to R\) is a counit
- \(\Delta: G \to G \otimes_R G\) 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 \(a\in H\)에 대하여,

\[a.(v\otimes w)= \Delta(a)(v\otimes w)\]

dual representation \(a\in H, f\in V^{*}\)에 대하여 \(a.f\)를 다음과 같이 정의

\[(a.f)(v)= f(S (a).v)\]

the category of representations has a monoidal structure with duals

예

group ring

유한군의 group algebra
\(H=\mathbb{F}G\) : group algebra of G over F
곱셈과 항등원

\[m : \mathbb Z[G]\otimes \mathbb Z[G] \to \mathbb Z[G]\]

comultiplication

\[\Delta : \mathbb Z[G] \to \mathbb Z[G]\otimes \mathbb Z[G] \] \[g \mapsto g\otimes g\]

counit\[\epsilon(g)=1\]
antipode\[S(g)=g^{-1}\]

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

사전 형태의 자료

리뷰, 에세이, 강의노트

Darij Grinberg, Victor Reiner, Hopf Algebras in Combinatorics, arXiv:1409.8356 [math.CO], September 30 2014, http://arxiv.org/abs/1409.8356

"호프 대수(Hopf algebra)"의 두 판 사이의 차이

2021년 2월 17일 (수) 03:50 기준 최신판

목차

개요

군(group) 의 정의 : abstract nonsense

호프 대수(Hopf algebra) 의 정의

표현론에서 유용한 점

예

group ring

UEA

역사

메모

관련된 항목들

사전 형태의 자료

관련도서

리뷰, 에세이, 강의노트

메타데이터

위키데이터

Spacy 패턴 목록

둘러보기 메뉴

검색

@@ 1번째 줄: / 1번째 줄: @@
-<h5>이 항목의 수학노트 원문주소</h5>
+==개요==
-* [[search?q=%ED%98%B8%ED%94%84%20%EB%8C%80%EC%88%98&parent id=12529068|호프 대수]]
+* 호프 대수(Hopf algebra) = bi-algebra with an antipode
+* '군 (group)' ([[군론(group theory)]] 항목 참조) 개념의 일반화
+*  양자군의 이론에서 중요한 역할
-<h5>개요</h5>
-* 호프 대수(Hopf algebra) = bi-algebra with an antipoe
-* '군(group)' ([[군론(group theory)]] 항목 참조) 개념의 일반화
-*  양자군의 이론에서 중요한 역할<br>
 ** 양자군(quantum group) = non co-commutative quasi-triangular Hopf algebra
+==군(group) 의 정의 : abstract nonsense==
-<h5>군(group) 의 정의 : abstract nonsense</h5>
 * 군의 정의를 abstract nonsense를 사용하여 표현하기
-*  a group is a set <math>G&bg=ffffff&fg=000000&s=0</math> equipped with<br>
+*  a group is a set <math>G</math> equipped with
-** a multiplication map <math>\mu: G \otimes G \to G</math>
+** a multiplication map <math>\mu: G \otimes G \to G</math>
-** an inversion map <math>S: G \to G</math>
+** an inversion map <math>S: G \to G</math>
-** an identity element <math>1:+*+\to+G&bg=ffffff&fg=000000&s=0</math>, where <math>*&bg=ffffff&fg=000000&s=0</math> is a one point set
+** an identity element <math>1:*\to G</math>, where <math>*</math> is a one point set
-** <math>\epsilon:+G+\to+*&bg=ffffff&fg=000000&s=0</math>  (trivial representation, counit)
+** <math>\epsilon:G\to *</math>  (trivial representation, counit)
-** <math>\Delta: G \to G \otimes G</math>, diagonal map: <math>g \mapsto g\otimes g</math>
+** <math>\Delta: G \to G \otimes G</math>, diagonal map <math>g \mapsto g\otimes g</math>
-* 일반적으로 군을 정의할 때 드러나지 않는 <math>\epsilon:+G+\to+*&bg=ffffff&fg=000000&s=0</math> , <math>\Delta:+G+\to+G+\times+G&bg=ffffff&fg=000000&s=0</math>를 도입함으로써, 군을 abstract nonsense 만으로 표현할 수 있게 된다
+* 일반적으로 군을 정의할 때 드러나지 않는 <math>\epsilon:G \to *</math> , <math>\Delta:G \to G \times G</math>를 도입함으로써, 군을 abstract nonsense 만으로 표현할 수 있게 된다
-*  결합법칙<br><math>\mu \circ (\mu \otimes \operatorname{id})=\mu \circ (\operatorname{id}\otimes \mu)</math><br>
+*  결합법칙:<math>\mu \circ (\mu \otimes \operatorname{id})=\mu \circ (\operatorname{id}\otimes \mu)</math>
-*  역원에 대한 조건 (원소에 그 역원을 곱하면 항등원을 얻는다)<br><math>\mu \circ (\operatorname{id}\otimes S) \circ \Delta = \mu \circ (S\otimes \operatorname{id}) \circ \Delta=1 \circ \epsilon</math>, i.e., multiplying an element with its inverse yields the unit.<br>
+*  역원에 대한 조건 (원소에 그 역원을 곱하면 항등원을 얻는다):<math>\mu \circ (\operatorname{id}\otimes S) \circ \Delta = \mu \circ (S\otimes \operatorname{id}) \circ \Delta=1 \circ \epsilon</math>
+==호프 대수(Hopf algebra) 의 정의==
-<h5>호프 대수(Hopf algebra) 의 정의</h5>
+* Hopf algebra = an algebra with unit and three maps = bialgebra with an antipode
+*  Given a commutative ring <math>R</math>, a Hopf algebra over <math>R</math> is a six-tuple <math>(G, \mu, 1, S, \epsilon, \Delta)</math>,
-* Hopf algebra = an algebra with unit and three maps = bialgebra with an antipode
+** <math>G</math>is an <math>R</math>-module
-*  Given a commutative ring <math>R&bg=ffffff&fg=000000&s=0</math>, a Hopf algebra over <math>R&bg=ffffff&fg=000000&s=0</math> is a six-tuple <math>(G, \mu, 1, S, \epsilon, \Delta)</math>,<br>
+** <math>\mu: G \otimes_R G \to G</math> is a multiplication map
-** <math>G&bg=ffffff&fg=000000&s=0</math>is an <math>R&bg=ffffff&fg=000000&s=0</math>-module
+** <math>1:R \to G</math> is a unit
-** <math>\mu: G \otimes_R G \to G</math> is a multiplication map
+** <math>S: G \to G</math> is called the antipode
-** <math>1:+R+\to+G&bg=ffffff&fg=000000&s=0</math> is a unit
+** <math>\epsilon: G \to R</math> is a counit
-** <math>S: G \to G</math> is called the antipode
+** <math>\Delta: G \to G \otimes_R G</math> is called comultiplication.
-** <math>\epsilon:+G+\to+R&bg=ffffff&fg=000000&s=0</math> is a counit
+*  These are required to satisfy relations
-** <math>\Delta:+G+\to+G+\otimes_R+G&bg=ffffff&fg=000000&s=0</math> is called comultiplication.
+** <math>(G,\mu,1)</math>  ring
-*  These are required to satisfy relations<br>
-** <math>(G,\mu,1)</math>  ring
 ** <math>(G,\Delta,\epsilon)</math> coring (just turn all the previous arrows around)
 ** comultiplication and counit are a ring 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>
-<h5>표현론에서 유용한 점</h5>
+==표현론에서 유용한 점==
 * H : Hopf algebra
 * V,W : H-modules
 * 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
-*  tensor product<br><math>a.(v\otimes w)= \Delta(a)(v\otimes w)</math><br>
+* counit - trivial representations
-*  dual representation<br> For <math>f\in V^{*}</math>, <math>(a.f)(v)= f(S(a).v)</math><br>
+* tensor product <math>a\in H</math>에 대하여,
+:<math>a.(v\otimes w)= \Delta(a)(v\otimes w)</math>
+* dual representation  <math>a\in H, f\in V^{*}</math>에 대하여 <math>a.f</math>를 다음과 같이 정의
+:<math>(a.f)(v)= f(S (a).v)</math>
+* the category of representations has a monoidal structure with duals
-<h5>예 : group ring</h5>
-<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>
+==예==
-*  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>
+===group ring===
-*  counit<br><math>\epsilon(g)=1</math><br>
+* [[유한군의 group algebra]]
-*  antipode<br><math>S(g)=g^{-1}</math><br>
+* <math>H=\mathbb{F}G</math> : group algebra of G over F
+* 곱셈과 항등원
+:<math>m : \mathbb Z[G]\otimes \mathbb Z[G] \to \mathbb Z[G]</math>
+* comultiplication
+:<math>\Delta : \mathbb Z[G] \to \mathbb Z[G]\otimes \mathbb Z[G] </math>
+:<math>g \mapsto g\otimes g</math>
+* counit:<math>\epsilon(g)=1</math>
+* antipode:<math>S(g)=g^{-1}</math>
-<h5>예 : UEA</h5>
+===UEA===
+* simple Lie algebra <math>\mathfrak{g}</math>
-* simple Lie algebra g
 * <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):<math>\Delta : U (\mathfrak{g}) \to U (\mathfrak{g})\otimes U(\mathfrak{g}) </math>:<math>\Delta(x) =x\otimes 1+ 1 \otimes x</math> for <math>x \in \mathfrak{g}</math>:<math>\Delta(1)=1\otimes 1</math>
+*  counit:<math>\epsilon(x) =0</math> for <math>x \in \mathfrak{g}</math>:<math>\epsilon(1) =1</math>
-*  counit<br><math>\epsilon(x) =0</math> for <math>x \in \mathfrak{g}</math><br><math>\epsilon(1) =1</math><br>
+*  antipode:<math>S(x) = -x</math> for <math>x \in \mathfrak{g}</math>:<math>S(1) =1</math>
-*  antipode<br><math>S(x) = -x</math> for <math>x \in \mathfrak{g}</math><br><math>S(1) =1</math><br>
+* [[quantized universal enveloping algebra]]
-* [[quantized universal enveloping algebra]]<br>
-<h5>역사</h5>
+==역사==
 * http://www.google.com/search?hl=en&tbs=tl:1&q=
-* [[수학사연표 (역사)|수학사연표]]
+* [[수학사 연표]]
-<h5>메모</h5>
-* Math Overflow http://mathoverflow.net/search?q=
-<h5>관련된 항목들</h5>
-<h5>수학용어번역</h5>
+==메모==
-*  단어사전<br>
+* [http://sbseminar.wordpress.com/2007/10/07/group-hopf-algebra/ Group = Hopf algebra] , Scott Carnahan, Secret Blogging Seminar,  October 7, 2007
-** http://translate.google.com/#en|ko|
+* http://mathoverflow.net/questions/101739/is-there-any-progress-on-the-theory-in-the-paper-geometric-methods-in-representa
-** http://ko.wiktionary.org/wiki/
-* 발음사전 http://www.forvo.com/search/
-* [http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=&fstr= 대한수학회 수학 학술 용어집]<br>
-** http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=eng_term&fstr=
-* [http://www.kss.or.kr/pds/sec/dic.aspx 한국통계학회 통계학 용어 온라인 대조표]
-* [http://cgi.postech.ac.kr/cgi-bin/cgiwrap/sand/terms/terms.cgi 한국물리학회 물리학 용어집 검색기]
-* [http://www.nktech.net/science/term/term_l.jsp?l_mode=cate&s_code_cd=MA 남·북한수학용어비교]
-* [http://kms.or.kr/home/kor/board/bulletin_list_subject.asp?bulletinid=%7BD6048897-56F9-43D7-8BB6-50B362D1243A%7D&boardname=%BC%F6%C7%D0%BF%EB%BE%EE%C5%E4%B7%D0%B9%E6&globalmenu=7&localmenu=4 대한수학회 수학용어한글화 게시판]
-<h5>매스매티카 파일 및 계산 리소스</h5>
+==관련된 항목들==
+* [[공대수 (coalgebra)]]
-*
-* http://www.wolframalpha.com/input/?i=
-* http://functions.wolfram.com/
-* [http://dlmf.nist.gov/ NIST Digital Library of Mathematical Functions]
-* [http://people.math.sfu.ca/%7Ecbm/aands/toc.htm Abramowitz and Stegun Handbook of mathematical functions]
-* [http://www.research.att.com/%7Enjas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]
-* [http://numbers.computation.free.fr/Constants/constants.html Numbers, constants and computation]
-* [https://docs.google.com/open?id=0B8XXo8Tve1cxMWI0NzNjYWUtNmIwZi00YzhkLTkzNzQtMDMwYmVmYmIxNmIw 매스매티카 파일 목록]
-<h5>사전 형태의 자료</h5>
+==사전 형태의 자료==
-* [http://ko.wikipedia.org/wiki/%ED%98%B8%ED%94%84_%EB%8C%80%EC%88%98 http://ko.wikipedia.org/wiki/호프_대수]
+* http://ko.wikipedia.org/wiki/호프_대수
-* [http://en.wikipedia.org/wiki/Hopf_algebra ]http://en.wikipedia.org/wiki/Hopf_algebra
+* http://en.wikipedia.org/wiki/Hopf_algebra
 * http://en.wikipedia.org/wiki/Coalgebra
-* [http://www.encyclopediaofmath.org/index.php/Main_Page Encyclopaedia of Mathematics]
+==관련도서==
-* [http://dlmf.nist.gov NIST Digital Library of Mathematical Functions]
+* Hazewinkel, Michiel, Nadezhda Mikhaĭlovna Gubareni, and Vladimir V. Kirichenko. 2010. Algebras, Rings, and Modules: Lie Algebras and Hopf Algebras. American Mathematical Soc. http://books.google.de/books?id=Q5K3vREGVhAC
-* [http://eqworld.ipmnet.ru/ The World of Mathematical Equations]
-<h5>리뷰논문, 에세이, 강의노트</h5>
-<h5>관련논문</h5>
-* http://www.jstor.org/action/doBasicSearch?Query=
-* http://www.ams.org/mathscinet
-* http://dx.doi.org/
-<h5>관련도서</h5>
+==리뷰, 에세이, 강의노트==
+* Darij Grinberg, Victor Reiner, Hopf Algebras in Combinatorics, arXiv:1409.8356 [math.CO], September 30 2014, http://arxiv.org/abs/1409.8356
-*  도서내검색<br>
+==메타데이터==
-** http://books.google.com/books?q=
+===위키데이터===
-** http://book.daum.net/search/contentSearch.do?query=
+* ID :  [https://www.wikidata.org/wiki/Q1627597 Q1627597]
+===Spacy 패턴 목록===
+* [{'LOWER': 'hopf'}, {'LEMMA': 'algebra'}]