# 호프 대수(Hopf algebra)

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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

## 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