User:Hillgentleman/fibre functor

設

• k - 係一域(field)
• $C$ - 係對稱張量範疇(symmetric tensor category)
• $Vect(k)$ - 係k上向量空間所組成嘅範疇

定義：

• 纖維函子(fibre functor) $F:C\longrightarrow Vect(k)$係
  • 一正則忠實辮狀張量函子( exact faithful braided tensor functor)，由範疇$C$射向$Vect(k)$。

A general proposition

Let

• k - algebrically closed field, with characteristic 0
• C - symmetric tensor category over k
  • - which is an abelian category
  • - which has finitely many indecomposable objects
• F : C ---> Vect(k) - fibre functor
• $G=Aut^{\otimes }(F)$
  • - the group of tensor automorphismata of F

Then

• G is a finite group
• C is equivalent to Rep(G), the category of representations of G.

references

• P.Etingof / O.Schiffmann (1998): "Lectures on Quantum Groups", International Press, ISBN 1571460632, p.163
• Deligne / Milne (1982): "Tanakian Categories", Lecture Notes in Mathematics 900, [1], p.148