Given an abelian 𝑉-linear rigid monoidal category 𝑉, where 𝑉 is a perfect field, we define squared coalgebras as objects of cocompleted 𝑉 ⨂ 𝑉 (Deligne's tensor product of categories) equipped with the appropriate notion of comultiplication. Based on this, (squared) bialgebras and Hopf algebras are defined without use of braiding. If 𝑉 is the category of 𝑉-vector spaces, squared (co)algebras coincide with conventional ones. If 𝑉 is braided, a braided Hopf algebra can be obtained from a squared one. Reconstruction theorems give equivalence of squared co- (bi-, Hopf) algebras in 𝑉 and corresponding fibre functors to 𝑉 (which is not the case with the usual definitions). Finally, squared quasitriangular Hopf coalgebra is a solution to the problem of defining quantum groups in braided categories.
@article{bwmeta1.element.bwnjournal-article-bcpv40z1p111bwm,
author = {Lyubashenko, Volodymyr},
title = {Squared Hopf algebras and reconstruction theorems},
journal = {Banach Center Publications},
volume = {38},
year = {1997},
pages = {111-137},
zbl = {0904.16017},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-bcpv40z1p111bwm}
}
Lyubashenko, Volodymyr. Squared Hopf algebras and reconstruction theorems. Banach Center Publications, Tome 38 (1997) pp. 111-137. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-bcpv40z1p111bwm/
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