호프 대수(Hopf algebra)

이 항목의 수학노트 원문주소

• 호프 대수

 

 

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 \(\mu \circ (\operatorname{id}\otimes S) \circ \Delta = \mu \circ (S\otimes \operatorname{id}) \circ \Delta=1 \circ \epsilon\)

 

 

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
• tensor product
  \(a.(v\otimes w)= \Delta(a)(v\otimes w)\)
  
• dual representation
  For \(f\in V^{*}\), \((a.f)(v)= f(S(a).v)\)
  

 

 

 

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 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

 

• http://www.google.com/search?hl=en&tbs=tl:1&q=
• 수학사연표

 

 

메모

 

• Math Overflow http://mathoverflow.net/search?q=

 

 

 

 

수학용어번역

• 단어사전
  
  • http://translate.google.com/#en%7Cko%7C
  • http://ko.wiktionary.org/wiki/
• 발음사전 http://www.forvo.com/search/
• 대한수학회 수학 학술 용어집
  
  • http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=eng_term&fstr=
• 한국통계학회 통계학 용어 온라인 대조표
• 한국물리학회 물리학 용어집 검색기
• 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

 

 

 

 

 

관련논문

• http://www.jstor.org/action/doBasicSearch?Query=
• http://www.ams.org/mathscinet
• http://dx.doi.org/

 

 

관련도서

• 도서내검색
  
  • http://books.google.com/books?q=
  • http://book.daum.net/search/contentSearch.do?query=