Eulerian idempotents and Milnor-Moore theorem for certain non-cocommutative Hopf algebras
type: Item
instance of: scholarly article
title: Eulerian idempotents and Milnor-Moore theorem for certain non-cocommutative Hopf algebras
zbMATH Open document ID: 1017.16033
DOI: 10.1016/S0021-8693(02)00097-2
published in: Journal of Algebra
publication date: 2003-02-12T00:00:00Z
review text: A classical result of \textit{J. W. Milnor} and \textit{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.
reviewed by: Jörg Feldvoss
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 16W50
Mathematics Subject Classification ID: 16S30
zbMATH DE Number: 1867974
zbMATH Keywords: connected graded cocommutative Hopf algebras
zbMATH Keywords: universal enveloping algebras
zbMATH Keywords: Lie algebras of primitive elements
zbMATH Keywords: dendriform Hopf algebras
zbMATH Keywords: brace algebras
zbMATH Keywords: Eulerian projections
author: María O. Ronco
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1867974