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On Hopf algebras and rigid monoidal categories. (English) Zbl 0727.16029

If \({\mathcal C}\) is a k-linear, Abelian category over a field k and w: \({\mathcal C}\to Vec_ f(k)\) is a k-linear exact and faithful functor into the category of finite-dimensional vector spaces, there is a k-coalgebra A such that \({\mathcal C}\) is k-linearly equivalent to \(Comod_ f(A)\), the category of finite-dimensional right A-comodules [N. Saavedra Rivano, Catégories Tannakiennes (Lect. Notes Math. 265, 1972; Zbl 0241.14008)]. If \({\mathcal C}\) has a rigid symmetric monoidal structure (i.e. is symmetric monoidal with a duality) then A is a commutative Hopf algebra. In this note, it is shown that there is still a Hopf algebra structure on A if one removes the symmetricity requirement on the monoidal structure.
Reviewer: T.Porter (Bangor)

MSC:

16W30 Hopf algebras (associative rings and algebras) (MSC2000)
18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
19D23 Symmetric monoidal categories

Citations:

Zbl 0241.14008
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References:

[1] P. Deligne,Catégories Tannakiennes, preprint.
[2] P. Deligne and J. Milne,Tannakian Categories, Lecture Notes in Math.900, Springer-Verlag, Berlin, 1982, pp. 101–228. · Zbl 0477.14004
[3] N. Saavedra Rivano,Catégories Tannakiennes, Lecture Notes in Math.265, Springer-Verlag, Berlin, 1972. · Zbl 0241.14008
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