"호프 대수(Hopf algebra)"의 두 판 사이의 차이

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2012년 11월 2일 (금) 08:40 판

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 \(G\) 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)
    \(\mu \circ (\operatorname{id}\otimes S) \circ \Delta = \mu \circ (S\otimes \operatorname{id}) \circ \Delta=1 \circ \epsilon\), 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 \(R\), 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
    \(a.(v\otimes w)= \Delta(a)(v\otimes w)\)
  • dual representation
    For \(f\in V^{*}\), \((a.f)(v)= f(S (a).v)\)
  • the category of representations has a monoidal structure with duals



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 \(\mathfrak{g}\)
  • \(U(\mathfrak{g})\)
  • 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 \(x \in \mathfrak{g}\)
    \(\Delta(1)=1\otimes 1\)






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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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