On homomorphisms of cocommutative coalgebras and Hopf algebras (Q1094512)

Let \(k\to G\to^{j}H\to^{\rho}J\to k\) be an exact sequence of cocommutative Hopf algebras over a field k and let C be a cocommutative coalgebra over k. It is well-known that the induced sequence of groups \[ \{e\}\to Coalg(C,G)\to^{j_*}Coalg(C,H)\to^{\rho_*}Coalg(C,J) \] is also exact. \textit{T. Shudo} has shown that if C is also pointed and if H is a hyperalgebra, then \(\rho_*\) is surjective if and only if j has a coalgebra retraction [Hiroshima Math. J. 13, 627-646 (1983; Zbl 0529.16005)]. The author generalizes this result for cocommutative coalgebras C and for exact sequences of pointed cocommutative Hopf algebras, also showing that the condition is equivalent to \(\rho\) having a coalgebra splitting (as did Shudo for hyperalgebras). His methods involve reducing to group-like elements and nearly-primitive elements. Some of the preliminary results are interesting in themselves, but the reader should compare with the (not necessarily cocommutative) results for group-like elements and nearly-primitive elements in pointed coalgebras obtained by \textit{T. Marlowe} [J. Pure Appl. Algebra 35, 157- 169 (1985; Zbl 0598.57031)].