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Squared Hopf algebras. (English) Zbl 1230.18001

Mem. Am. Math. Soc. 677, x, 180 p. (1999).
The author introduces the principally new concept of squared coalgebra. Based on this, (squared) bialgebras and Hopf algebras are defined without use of braiding. These concepts allow one to recover the original form of the classical reconstruction theorem for the case of an abelian monoidal base category \(\mathcal V\), and to prove the equivalence of squared co- (bi-, Hopf) algebras in \(\mathcal V\) and corresponding fibre functors to \(\mathcal V\). A squared quasitriangular Hopf coalgebra is a solution to the problem of defining quantum groups in braided categories.
Let \(K\) be a perfect field. The classical reconstruction theorem asserts that a \(K\)-coalgebra can be reconstructed from the underlying functor from its category of comodules to vector spaces. Saavedra (1972) and later Schauenburg (1992) proved that an essentially small abelian \(K\)-linear category \(\mathcal C\) equipped with an exact faithful functor \(\omega: \mathcal C\to K\)-vect to the category of finite-dimensional \(K\)-vector spaces is equivalent to the category of finite-dimensional comodules over some \(K\)-coalgebra, and the initial functor \(\omega\) is the composite of this equivalence \(F\) and the underlying functor to vector spaces. When \(K\)-vect is replaced by an abelian \(K\)-linear rigid monoidal category \(\mathcal V\), a version of the reconstruction theorem holds with the coalgebra \(\overline C\) reconstructed by the coend \(\int^{X\in{\mathcal C}} \omega(X) \otimes_{K}\omega(X)^*\)in cocompletion \(\widehat{\mathcal V}\), although \(F\) is no longer an equivalence.
It is the central observation of the paper that by modifying the definitions of coalgebras and comodules one can make \(F\) into an equivalence. Namely, instead of the coalgebra \(\overline C\) in \(\widehat{\mathcal V}\) one can use the squared coalgebra \(C=\int^{X\in{\mathcal C}}\omega(X)\boxtimes_{K}\omega(X)^*\) in \(\widehat{{\mathcal V}\boxtimes{\mathcal V}}\). The notion of squared coalgebra is based on Deligne’s tensor product \(\boxtimes\) of abelian categories. A squared coalgebra is an object of the cocompleted Deligne tensor square \(\widehat{{\mathcal V}^{\boxtimes 2}}\) of the initial category \(\mathcal V\) (whence the terminology) equipped with the appropriate notion of comultiplication. Thus, the full form of the reconstruction theorem asserts equivalence of the following two categories: the category of \(K\)-linear exact faithful functors from an essentially small category to \(\mathcal V\) and the category of squared coalgebras in \(\mathcal V\). The internal tensor product functor \(\otimes: \mathcal V\times\mathcal V \to \mathcal V\) can be decomposed as \(\mathcal V\times\mathcal V \buildrel\boxtimes\over\rightarrow \mathcal V\boxtimes\mathcal V \buildrel\circledast\over\rightarrow\mathcal V\) and, hence, the ordinary coalgebra (comonoid) is in the form \(\overline C = \circledast C\). The author shows that the corresponding category of comodules \({}^{\overline C}\mathcal V\) is equivalent to \({}^C{\mathcal V} \boxtimes{\mathcal V}\). This explains the appearance of “hidden symmetries” in the coend reconstructed from the comodule category over braided Hopf algebra (Pareigis, 1996). The monoidal version of the reconstruction theorem dictates the definition of a squared bialgebra. Squared Hopf (co)algebras based on \(\mathcal V\) can also be defined, even if \(\mathcal V\) is not braided but satisfies a much weaker condition. If \(\mathcal V\) is the category of \(K\)-vector spaces, squared (co)algebras coincide with conventional ones.
If \(\mathcal V\) is braided, a squared Hopf (co)algebra determines a braided Hopf algebra, but not vice versa. In this case, the author proves the equivalence of the category of monoidal \(K\)-linear exact faithful functors \(\omega: {\mathcal C}\to{\mathcal V}\), where \(\mathcal C\) is rigid braided, and of the category of quasitriangular Hopf coalgebras, appropriately defined. In particular, the category of comodules over a quasitriangular Hopf coalgebra is braided. This is not trivial, and allows the author to introduce a non-obvious braiding for the bigger category of comodules over the braided Hopf algebra \(\circledast H\). However, it seems impossible, in general, to introduce a braided structure of any kind for the whole category of comodules over a braided Hopf algebra not related with Hopf coalgebras. Thus the notion of a quasitriangular Hopf coalgebra is the closest to the idea of a “quantum group in a braided category”. Similar notions exist for ribbon categories.
The author pays some attention also to the background on which the action develops – the higher category theory. Besides the usual definitions of monoidal, braided and symmetric categories a version of these definitions with \(n\)-ary tensor operations is given. The author proves that the 2-category of such categories is 2-equivalent to its conventional counterpart. This fits into the general picture of Batanin, who shows that the definition of a weak \(m\)-category is not unique, but unique up to an equivalence. The chosen definition allows the author to prove that Deligne’s tensor product gives indeed a weak symmetric monoidal structure on a 2-category of abelian categories.

MSC:

18-02 Research exposition (monographs, survey articles) pertaining to category theory
18D15 Closed categories (closed monoidal and Cartesian closed categories, etc.)
16T05 Hopf algebras and their applications
18D20 Enriched categories (over closed or monoidal categories)
18E10 Abelian categories, Grothendieck categories
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