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Indecomposable coalgebras, simple comodules, and pointed Hopf algebras. (English) Zbl 0836.16024

Let \(S\) be the set of simple subcoalgebras of a coalgebra \(C\). The elements of \(S\) are the vertices of the quiver \(\Gamma_C\). Two vertices \(S_1\), \(S_2\) are linked by an edge \(S_1 \to S_2\) if \(S_1 + S_2 \neq \Delta^{-1} (S_1 \otimes S_2 + S_2 \otimes S_1)\). \(\Gamma_C\) as a directed graph is isomorphic to the Ext quiver of simple (right) \(C\)-comodules. \(\Gamma_C\) is connected if and only if \(C\) is indecomposable. Let \(H\) be a pointed Hopf algebra. Denote by \(H_{(1)}\) the indecomposable component containing the unit element. Then \(H_{(1)}\) is a Hopf subalgebra, \(G(H_{(1)})\) is a normal subgroup in \(G(H)\) and \(H\) is a crossed product of \(H_{(1)}\) and the group algebra \(k(G(H)/G(H_{(1)}))\).
Similar results were considered by I. Kaplansky [Bialgebras (Lect. Notes, Chicago Univ., 1975)], T. Shudo and H. Miyamoto [Hiroshima Math. J. 8, 499-504 (1978; Zbl 0412.16005)], R. G. Heyneman and M. E. Sweedler [J. Algebra 13, 192-241 (1969; Zbl 0203.31601)].

MSC:

16W30 Hopf algebras (associative rings and algebras) (MSC2000)
16S35 Twisted and skew group rings, crossed products
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