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From left modules to algebras over an operad: application to combinatorial Hopf algebras. (English) Zbl 1206.18010

An \(\mathbb S\)-module is a graded vector space \((V_n)_{n\geq 0}\) together with a right action of the symmetric group \(S_n\) on \(V_n\) for each \(n\). Let \(\mathcal O\) be the forgetful functor from \(\mathbb S\)-modules to graded vector spaces. Given an operad \(\mathcal P\), the following notions are the same: twisted \(\mathcal P\)-algebras, left modules over \(\mathcal P\), \(\mathcal P\)-algebras in the category of \(\mathbb S\)-modules. The first question studied in the paper under review is the following: given a \(\mathcal P\)-algebra \(M\) in the category S-mod, how to endow the graded vector space \({\mathcal O}(M)\) with a \(\mathcal P\)-algebra structure?
It is inspired by the observation that if \(\oplus_n{\mathcal P}(n)\) is a \(\mathcal P\)-algebra in S-mod, it is not obligatory a \(\mathcal P\)-algebra in the category of graded vector spaces. The author shows that if one applies a symmetrization to the twisted \(\mathcal P\)-algebra structure on \(M\), then \({\mathcal O}(M)\) is a \(\mathcal P\)-algebra. If the operad \(\mathcal P\) is regular, then he defines another product. In the special case when \({\mathcal P}={\mathcal A}s\) is the associative operad, the constructions recover constructions of F. Patras and C. Reutenauer [Mosc. Math. J. 4, No. 1, 199–216 (2004; Zbl 1103.16026)]. Then the author defines the notion of Hopf \(\mathcal P\)-algebras in the category S-mod which, for \({\mathcal P}={\mathcal A}s\), recovers the notion of twisted associative bialgebras.
Again, the functor \(\mathcal O\) sends a Hopf \(\mathcal P\)-algebra to a Hopf \(\mathcal P\)-algebra. Also, another construction is introduced in the regular case. For \(\mathcal P\) regular the author defines unital infinitesimal \(\mathcal P\)-bialgebras which, restricted to \({\mathcal P}={\mathcal A}s\), give a construction of J.-L. Loday and M. Ronco [Contemp. Math. 346, 369–398 (2004; Zbl 1065.18007)]. Combined with results of Loday and Ronco, this implies freeness and cofreeness results for Hopf algebras built from Hopf operads.
Finally, the author proves that many combinatorial Hopf algebras arise from his theory, as it is the case for various Hopf algebras defined on the faces of the permutohedra and associahedra.

MSC:

18D50 Operads (MSC2010)
16T05 Hopf algebras and their applications
16T10 Bialgebras
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References:

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