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Descent algebra and cogroups in algebras over an operad. (Algèbre des descentes et cogroupes dans les algèbres sur une opérade.) (French) Zbl 0940.18004

The author gives a new proof of his theorem that cogroups in algebras over an operad \(\mathcal{P}\) are free \(\mathcal{P}\)-algebras [B. Fresse, Comment. Math. Helv. 73, No. 4, 637-676 (1998; Zbl 0929.16033)]. By algebra is meant either a connected graded algebra or a complete algebra. The ground field has to be of characteristic \(0\). The proof relies on a universal action of Solomon’s descent algebra (for symmetric groups) [L. Solomon, J. Algebra 9, 220-239 (1968; Zbl 0186.04503)] on \(\mathcal{P}\)-algebras with cogroup structure. The action of the first Eulerian idempotent induces moreover an explicit set of free generators. In the special case of connected commutative Hopf algebras the same construction has been carried out by F. Patras [Ann. Inst. Fourier 43, No. 4, 1067-1087 (1993; Zbl 0795.16028)]. In the case of a connected differential graded \(\mathcal{P}\)-algebra with cogroup structure, the universal action is compatible with the differential. By duality, this implies that for a connected graded \(\mathcal{P}\)-coalgebra with group structure, the homology of its primitive part is naturally isomorphic to the primitive part of its homology.

MSC:

18D50 Operads (MSC2010)
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
16W50 Graded rings and modules (associative rings and algebras)
05E10 Combinatorial aspects of representation theory
18D35 Structured objects in a category (MSC2010)
20F29 Representations of groups as automorphism groups of algebraic systems
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