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Coherent unit actions on regular operads and Hopf algebras

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Kurusch Ebrahimi-Fard and Li Guo

J.-L. Loday introduced the concept of coherent unit actions on a
regular operad and showed that such actions give
Hopf algebra structures on the free algebras. Hopf algebras
obtained this way include the Hopf algebras of shuffles,
quasi-shuffles and planar rooted trees. We characterize coherent
unit actions on binary quadratic regular operads in terms of
linear equations of the generators of the operads. We then use
these equations to classify operads with coherent unit actions. We
further show that coherent unit actions are preserved under taking
products and thus yield Hopf algebras on the free object of the
product operads when the factor operads have coherent unit
actions. On the other hand, coherent unit actions are never
preserved under taking the dual in the operadic sense except for
the operad of associative algebras.

Keywords:
dendriform algebras, coherent unit actions, regular operads, Hopf algebras

2000 MSC:
18D50, 17A30, 16W30

*Theory and Applications of Categories,*
Vol. 18, 2007,
No. 13, pp 348-371.

http://www.tac.mta.ca/tac/volumes/18/13/18-13.dvi

http://www.tac.mta.ca/tac/volumes/18/13/18-13.ps

http://www.tac.mta.ca/tac/volumes/18/13/18-13.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/18/13/18-13.dvi

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/18/13/18-13.ps

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