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On the indecomposable elements of the bar construction. (English) Zbl 0613.55007

Let E be an augmented commutative differential graded algebra over a field k of characteristic zero. Then, the bar construction B(E) on E is a commutative differential Hopf algebra with augmentation ideal I and \(B(E)=k\oplus I\). The author proves the existence and gives an explicit formula of a splitting s: I/I\({}^ 2\to I\) of the natural projection. Further, s induces an isomorphism from the free commutative differential graded algebra generated by \(I/I^ 2\) into B(E), which may be seen as a dual of the Poincaré-Birkhoff-Witt theorem.
Reviewer: W.Grölz

MSC:

55P62 Rational homotopy theory
57T05 Hopf algebras (aspects of homology and homotopy of topological groups)
16W50 Graded rings and modules (associative rings and algebras)
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References:

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