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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.

##### MSC:
 16W30 Hopf algebras (associative rings and algebras) (MSC2000) 16W50 Graded rings and modules (associative rings and algebras) 16S30 Universal enveloping algebras of Lie algebras
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