## 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

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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