Hokkaido Mathematical Journal

ISHIDA Tomohiko, KAWAZUMI Nariya,

The Lie algebra of rooted planar trees.

Hokkaido Mathematical Journal, 42 (2013) pp.397-416

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Abstract

We study a natural Lie algebra structure on the free vector space generated by all rooted planar trees as the associated Lie algebra of the nonsymmetric operad (non-Σ operad, preoperad) of rooted planar trees. We determine whether the Lie algebra and some related Lie algebras are finitely generated or not, and prove that a natural surjection called the augmentation homomorphism onto the Lie algebra of polynomial vector fields on the line has no splitting preserving the units.

MSC(Primary)18D50
MSC(Secondary)57R32
Uncontrolled Keywordsnonsymmetric operad, polynomial vector field