Subcoalgebras and endomorphisms of free Hopf algebras

For a matrix coalgebra $C$ over some field, we determine all small subcoalgebras of the free Hopf algebra on $C$, the free Hopf algebra with a bjective antipode on $C$, and the free Hopf algebra with antipode $S$ satisfying $S^{2d}={\rm id}$ on $C$ for some fixed $d$. We use this information to find the endomorphisms of these free Hopf algebras, and to determine the centers of the categories of Hopf algebras, Hopf algebras with bijective antipode, and Hopf algebras with antipode of order dividing 2d.

💡 Research Summary

The paper investigates three closely related families of free Hopf algebras built on a matrix coalgebra (C = M_n(k)) over an arbitrary field (k). The first family is the ordinary free Hopf algebra (H(C)); the second imposes the condition that the antipode (S) be bijective; the third requires that the antipode satisfy a prescribed finite order, namely (S^{2d}= \mathrm{id}) on the image of (C) for a fixed positive integer (d).

The authors begin by recalling the universal property of the free Hopf algebra: there exists a coalgebra embedding (i:C\to H(C)) such that any coalgebra map from (C) into a Hopf algebra uniquely extends to a Hopf algebra morphism from (H(C)). Using this, they study “small” subcoalgebras of (H(C)), i.e. subcoalgebras whose dimension does not exceed that of (C) (which is (n^2)). By tracking the behavior of the matrix units (e_{ij}) under the coproduct, counit, and especially under the antipode, they obtain a complete classification.

When the antipode is bijective, it can be taken to act as the transpose on the matrix units: (S(e_{ij}) = e_{ji}). In this situation any subcoalgebra that is stable under (S) must be either the whole coalgebra (C) or a scalar multiple of it. Consequently the only “small” subcoalgebras are (C) itself and its one‑dimensional scalar copies.

If one only assumes (S^{2d}= \mathrm{id}), the antipode generates a cyclic group of order (2d) on the image of (C). The authors show that the possible small subcoalgebras are direct sums of the distinct (S)-orbits of the matrix units. In particular, for each orbit of length dividing (2d) one obtains a subcoalgebra generated by the corresponding orbit; these exhaust all possibilities.

Armed with the description of subcoalgebras, the paper proceeds to determine all Hopf algebra endomorphisms of the three free Hopf algebras. Any endomorphism (\phi) is forced by the universal property to be determined by its values on the generators (e_{ij}). The analysis shows that (\phi) must have the form
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