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Eulerian idempotents and Milnor-Moore theorem for certain non-cocommutative Hopf algebras. (English) Zbl 1017.16033
A classical result of J. W. Milnor and J. C. Moore [Ann. Math. (2) 81, 211-264 (1965; Zbl 0163.28202)] says that every connected graded cocommutative Hopf algebra over a field of characteristic zero is isomorphic to the universal enveloping algebra of its Lie algebra of primitive elements. The aim of the paper under review is to establish a similar result for a certain class of non-cocommutative Hopf algebras, namely the dendriform Hopf algebras.
A dendriform algebra is a (non-unital) associative algebra such that the product is the sum of two binary operations \(\prec\) and \(\succ\) satisfying certain conditions and a dendriform Hopf algebra has in addition a coproduct compatible with \(\prec\) and \(\succ\). A brace algebra is a vector space with a family of \(n\)-ary operations on itself satisfying certain conditions. Every dendriform algebra has the structure of a brace algebra. Moreover, the subspace of primitive elements (i.e., elements with zero coproduct) of a dendriform Hopf algebra is a brace subalgebra. Then the main result of this paper says that every connected graded dendriform Hopf algebra over a field of characteristic zero is isomorphic to the dendriform analogue of the universal enveloping algebra of its brace algebra of primitive elements. One ingredient of the proof is the construction of a family of Eulerian projections for connected graded dendriform Hopf algebras.

16W30 Hopf algebras (associative rings and algebras) (MSC2000)
16W50 Graded rings and modules (associative rings and algebras)
16S30 Universal enveloping algebras of Lie algebras
Full Text: DOI
[1] Barcucci, E.; Verri, C., Some more properties of Catalan numbers, Discrete math., 102, 229-237, (1992) · Zbl 0757.05005
[2] Chapoton, F., Un théorème de cartier – milnor – moore – quillen pour LES bigèbres dendriformes et LES algèbres braces · Zbl 0994.18006
[3] Gerstenhaber, M., The cohomology structure of an associative ring, Ann. of math., 78, 2, 267-288, (1963) · Zbl 0131.27302
[4] Gerstenhaber, M.; Shack, S., A Hodge-type decomposition for commutative algebra cohomology, J. pure appl. algebra, 48, 229-247, (1987) · Zbl 0671.13007
[5] Gerstenhaber, M.; Voronov, A., Homotopy G-algebras and moduli space operad, Internat. math. res. notices, 3, 141-153, (1995) · Zbl 0827.18004
[6] Getzler, E., Cartan homotopy formulas and the gauss – manin connection in cyclic homology, (), 65-78 · Zbl 0844.18007
[7] Kadeishvili, T., The structure of the A(∞)-algebra, and the Hochschild and Harrison cohomologies, (), 19-27 · Zbl 0717.55011
[8] Loday, J.-L., Opérations sur l’homologie cyclique des algèbres commutatives, Invent math., 96, 1, 205-230, (1989) · Zbl 0686.18006
[9] Loday, J.-L., Cyclic homology, Grundlehren math. wiss., 301, (1994), Springer-Verlag
[10] Loday, J.-L., Dialgebras, (), 7-66 · Zbl 0999.17002
[11] Milnor, J.W.; Moore, J.C., On the structure of Hopf algebras, Ann. of math., 81, 211-264, (1965) · Zbl 0163.28202
[12] Malvenuto, C.; Reutenauer, C., Duality between quasi-symmetric functions and the Solomon descent algebra, J. algebra, 177, 3, 967-982, (1995) · Zbl 0838.05100
[13] Ronco, M., Primitive elements in a free dendriform algebra, Contemp. math., (2000), to appear · Zbl 0974.16035
[14] Ronco, M., A milnor – moore theorem for dendriform Hopf algebras, C. R. acad. sci. Paris Sér. I, 332, 109-114, (2000) · Zbl 0978.16031
[15] Solomon, L., A MacKey formula in the group ring of a Coxeter group, J. algebra, 41, 2, 255-264, (1976)
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