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 \(G\) 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

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

  • 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



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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