호프 대수(Hopf algebra)

수학노트
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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 <math>G</math> equipped with
    • a multiplication map <math>\mu: G \otimes G \to G</math>
    • an inversion map <math>S: G \to G</math>
    • an identity element <math>1:*\to G</math>, where <math>*</math> is a one point set
    • <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>
  • 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:<math>\mu \circ (\mu \otimes \operatorname{id})=\mu \circ (\operatorname{id}\otimes \mu)</math>
  • 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

  • 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>,
    • <math>G</math>is an <math>R</math>-module
    • <math>\mu: G \otimes_R G \to G</math> is a multiplication map
    • <math>1:R \to G</math> is a unit
    • <math>S: G \to G</math> is called the antipode
    • <math>\epsilon: G \to R</math> is a counit
    • <math>\Delta: G \to G \otimes_R G</math> is called comultiplication.
  • These are required to satisfy relations
    • <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>, 많은 경우는 antihomomorphism 즉, <math>S(ab)=S(b)S(a)</math>


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
  • counit - trivial representations
  • 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


예

group ring

<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>


UEA

  • simple Lie algebra <math>\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):<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>
  • antipode:<math>S(x) = -x</math> for <math>x \in \mathfrak{g}</math>:<math>S(1) =1</math>
  • quantized universal enveloping algebra



역사



메모



관련된 항목들



사전 형태의 자료


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


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