In mathematics, a pre-Lie algebra is an algebraic structure on a vector space that describes some properties of objects such as rooted trees and vector fields on affine space.

The notion of pre-Lie algebra has been introduced by Murray Gerstenhaber in his work on deformations of algebras.

Pre-Lie algebras have been considered under some other names, among which one can cite left-symmetric algebras, right-symmetric algebras or Vinberg algebras.

Definition

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A pre-Lie algebra   is a vector space   with a linear map  , satisfying the relation  

This identity can be seen as the invariance of the associator   under the exchange of the two variables   and  .

Every associative algebra is hence also a pre-Lie algebra, as the associator vanishes identically. Although weaker than associativity, the defining relation of a pre-Lie algebra still implies that the commutator   is a Lie bracket. In particular, the Jacobi identity for the commutator follows from cycling the   terms in the defining relation for pre-Lie algebras, above.

Examples

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Vector fields on an affine space

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Let   be an open neighborhood of  , parameterised by variables  . Given vector fields  ,   we define  .

The difference between   and  , is   which is symmetric in   and  . Thus   defines a pre-Lie algebra structure.

Given a manifold   and homeomorphisms   from   to overlapping open neighborhoods of  , they each define a pre-Lie algebra structure   on vector fields defined on the overlap. Whilst   need not agree with  , their commutators do agree:  , the Lie bracket of   and  .

Rooted trees

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Let   be the free vector space spanned by all rooted trees.

One can introduce a bilinear product   on   as follows. Let   and   be two rooted trees.

 

where   is the rooted tree obtained by adding to the disjoint union of   and   an edge going from the vertex   of   to the root vertex of  .

Then   is a free pre-Lie algebra on one generator. More generally, the free pre-Lie algebra on any set of generators is constructed the same way from trees with each vertex labelled by one of the generators.

References

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  • Chapoton, F.; Livernet, M. (2001), "Pre-Lie algebras and the rooted trees operad", International Mathematics Research Notices, 2001 (8): 395–408, doi:10.1155/S1073792801000198, MR 1827084.
  • Szczesny, M. (2010), Pre-Lie algebras and incidence categories of colored rooted trees, vol. 1007, p. 4784, arXiv:1007.4784, Bibcode:2010arXiv1007.4784S.