Journal of Noncommutative Geometry

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Volume 2, Issue 2, 2008, pp. 195–214
DOI: 10.4171/JNCG/19

Published online: 2008-06-30

The necklace Lie coalgebra and renormalization algebras

Wee Liang Gan[1] and Travis Schedler[2]

(1) UC Riverside
(2) University of Chicago

We give a natural monomorphism from the necklace Lie coalgebra, defined for any quiver, to Connes and Kreimer’s Lie coalgebra of trees, and extend this to a map from a certain quiver-theoretic Hopf algebra to Connes and Kreimer’s renormalization Hopf algebra as well as to pre-Lie versions. These results are direct analogues of Turaev’s results in 2004, by replacing algebras of loops on surfaces with algebras of paths on quivers. We also factor the morphism through an algebra of chord diagrams and explain the geometric version. We then describe how all of the Hopf algebras are uniquely determined by the pre-Lie structures and discuss noncommutative versions of the Hopf algebras.

Keywords: Necklace, renormalization, Connes–Kreimer, pre-Lie, tree-structure

Gan Wee, Schedler Travis: The necklace Lie coalgebra and renormalization algebras. J. Noncommut. Geom. 2 (2008), 195-214. doi: 10.4171/JNCG/19