호프 대수(Hopf algebra)

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

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

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

관련된 항목들

• 공대수 (coalgebra)

사전 형태의 자료

• http://ko.wikipedia.org/wiki/호프_대수
• http://en.wikipedia.org/wiki/Hopf_algebra
• http://en.wikipedia.org/wiki/Coalgebra

관련도서

• 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

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

메타데이터

위키데이터

• ID : Q1627597

Spacy 패턴 목록

• [{'LOWER': 'hopf'}, {'LEMMA': 'algebra'}]