Homology, Homotopy and Applications

Volume 13 (2011)

Number 2

Lie coalgebras and rational homotopy theory, I: graph coalgebras

Pages: 263 – 292

DOI: https://dx.doi.org/10.4310/HHA.2011.v13.n2.a16


Dev Sinha (Department of Mathematics, University of Oregon, Eugene, Or., U.S.A.)

Benjamin Walter (Middle East Technical University, Northern Cyprus Campus, Güzelyurt, Mersin, Turkey)


We give a new presentation of the Lie cooperad as a quotient of the graph cooperad, a presentation which is not linearly dual to any of the standard presentations of the Lie operad. We use this presentation to explicitly compute duality between Lie algebras and coalgebras, to give a new presentation of Harrison homology, and to show that Lyndon words yield a canonical basis for cofree Lie coalgebras.


Lie coalgebras; rational homotopy theory; graph cohomology

2010 Mathematics Subject Classification

16E40, 55P48, 55P62

Published 25 January 2012