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

Fresse, Benôit

Bull. Soc. Math. Fr. 126, No. 3, 407-433 (1998).

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.

Reviewer: Clemens Berger (Nice)

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

Keywords:

cogroup; Solomon’s descent algebra; operad

Citations:

Zbl 0929.16033; Zbl 0186.04503; Zbl 0795.16028

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