호프 대수(Hopf algebra)

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

군의 정의를 abstract nonsense를 사용하여 표현하기
a group is a set \(G&bg=ffffff&fg=000000&s=0\) equipped with
- a multiplication map \(\mu: G \otimes G \to G\)
- an inversion map \(S: G \to G\)
- an identity element \(1:+*+\to+G&bg=ffffff&fg=000000&s=0\), where \(*&bg=ffffff&fg=000000&s=0\) is a one point set
- \(\epsilon:+G+\to+*&bg=ffffff&fg=000000&s=0\) (trivial representation, counit)
- \(\Delta: G \to G \otimes G\), diagonal map\[g \mapsto g\otimes g\]
일반적으로 군을 정의할 때 드러나지 않는 \(\epsilon:+G+\to+*&bg=ffffff&fg=000000&s=0\) , \(\Delta:+G+\to+G+\times+G&bg=ffffff&fg=000000&s=0\)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.

\(H=\mathbb{F}G\) : 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
\(S(g)=g^{-1}\)

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