"호프 대수(Hopf algebra)"의 두 판 사이의 차이

2012년 7월 28일 (토) 05:28 판

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.

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

@@ 37번째 줄: / 37번째 줄: @@
 <h5>호프 대수(Hopf algebra) 의 정의</h5>
+* Hopf algebra = an algebra with unit and three maps = bialgebra with an antipode
+*  Given a commutative ring <math>R&bg=ffffff&fg=000000&s=0</math>, a Hopf algebra over <math>R&bg=ffffff&fg=000000&s=0</math> is a six-tuple <math>(G, \mu, 1, S, \epsilon, \Delta)</math>,<br>
+** <math>G&bg=ffffff&fg=000000&s=0</math>is an <math>R&bg=ffffff&fg=000000&s=0</math>-module
+** <math>\mu: G \otimes_R G \to G</math> is a multiplication map
+** <math>1:+R+\to+G&bg=ffffff&fg=000000&s=0</math> is a unit
+** <math>S: G \to G</math> is called the antipode
+** <math>\epsilon:+G+\to+R&bg=ffffff&fg=000000&s=0</math> is a counit
+** <math>\Delta:+G+\to+G+\otimes_R+G&bg=ffffff&fg=000000&s=0</math> is called comultiplication.
+*  These are required to satisfy relations<br>
+** <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
+**