Cogroups in algebras over an operad are free algebras. (English) Zbl 0929.16033

Fresse, Benoît

Comment. Math. Helv. 73, No. 4, 637-676 (1998).

Let \(k\) be a field of characteristic 0. In what follows, all the structures are defined over \(k\).
The classical Leray theorem asserts that a graded connected commutative Hopf algebra \(H\) is a free commutative algebra [J. W. Milnor, J. C. Moore, Ann. Math., II. Ser. 81, 211-264 (1965; Zbl 0163.28202)]. An analogous statement holds if \(H\) is complete instead of graded connected. Dually, a connected graded cocommutative Hopf algebra is a cofree cocommutative coalgebra. Commutative Hopf algebras are cogroups in the category of commutative algebras, and the Leray theorem can be generalized to cogroups in various categories of (connected graded or complete) algebras, e.g. associative algebras [I. Berstein, Trans. Am. Math. Soc. 115, 257-269 (1965; Zbl 0134.42404)] or dual Leibniz algebras [J.-M. Oudom, Contemp. Math. 202, 115-135 (1997; Zbl 0880.17002)].
The purpose of this paper is to extend these theorems to a general class of algebras, namely to complete algebras over a unital operad \(\mathcal P\). In particular, the author proves that a complete algebra equipped with a cogroup structure over a unital operad \(\mathcal P\) is the completion of a free \(\mathcal P\)-algebra (Theorem 0.1).
In what follows, we write \(\widehat{\mathcal P}\)-algebras for complete \(\mathcal P\)-algebras. The idea of the proof is as follows. First of all, call \(\mathcal P\)-linear coalgebra a coalgebra in the (symmetric monoidal) category of right \(\mathcal P\)-modules. There exists a pair of contravariant adjoint functors relating \(\widehat{\mathcal P}\)-algebras and connected \(\mathcal P\)-linear cocommutative coalgebras (Theorem 2.6). This duality converts a cogroup object into a group object.
On the other hand, the author shows that the methods of F. Patras [J. Algebra 170, No. 2, 547-566 (1994; Zbl 0819.16033)] for studying classical Hopf algebras extend to Hopf algebras in any graded \(k\)-linear monoidal category (Theorem A.8). Since a group object in the category of \(\mathcal P\)-linear coalgebras is a Hopf algebra in the monoidal category of right \(\mathcal P\)-modules, Theorem 0.1 follows by duality.
Similar results and proofs hold in the graded connected case or for groups in categories of graded connected coalgebras over a unital operad.

Reviewer: Frederic Patras (Nice)

MSC:

16T05	Hopf algebras and their applications
17A30	Nonassociative algebras satisfying other identities
18D50	Operads (MSC2010)
18D35	Structured objects in a category (MSC2010)
18D10	Monoidal, symmetric monoidal and braided categories (MSC2010)

Keywords:

cogroups; operads; Leray theorem; connected graded cocommutative Hopf algebras; free algebras; complete algebras over operads; symmetric monoidal categories

Citations:

Zbl 0163.28202; Zbl 0134.42404; Zbl 0880.17002; Zbl 0819.16033