Pre-Lie algebras and the rooted trees operad

By Frédéric Chapoton, M. Livernet. 2000

463 citations recorded by the source corpus.

A Pre-Lie algebra is a vector space L endowed with a bilinear product * : L \times L to L satisfying the relation (x*y)*z-x*(y*z)= (x*z)*y-x*(z*y), for all x,y,z in L. We give an explicit combinatorial description in terms of rooted trees of the operad associated to this type of algebras and prove that it is a Koszul operad.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/69d6a08105167a5a713b3ac8c6e3fa0337e700fe

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Citations in the corpus

Renormalising SPDEs in regularity structures (2017, methodology): “e space rooted trees and the pre-Lie algebra given by vector ﬁelds on Rd. Indeed, one has ΥF[•y •• •] = ΥF[•] ⊳ΥF[ •• •] , (F⊳G) = F·DG. InthecaseofSPDEs,themorphismproperty isgivenbyCorollary4.15. In[CL01], a universal result has been proved, namely that the pre-Lie algebra of rooted trees is freel”
A unified algebraic approach to the classical Yang–Baxter equation (2007, background): “Left-symmetric algebras are a class of nonassociative algebras coming from the study of convex homogeneous cones, affine manifolds and affine structures on Lie groups, deformation of associative algebras [10–13] and then appear in many fields in mathematics and mathematical physics, such as complex ”
On the Lie Enveloping Algebra of a Post-Lie Algebra (2014, unknown): “The free pre-Lie algebra can be described as the space of non-planar rooted trees with product given by grafting of trees [8].”
A rigidity theorem for pre-Lie algebras (2006, methodology): “?>=< 89:; 2 If we compare with the operad PL based on rooted trees defining pre-Lie algebras in [3], this composition is a summand of the composition in PL.”
On spineless cacti, Deligne's conjecture and Connes-Kreimer's Hopf algebra (2003, methodology): “For rooted trees the composition defined in [5] tells us to cut off the branches, glue the second tree to the truncated tree and redistribute the branches.”
Monoidal Functors, Species, and Hopf Algebras (2010, background): “Some other related types of algebras prominent in the recent literature are diassociative algebras, preLie algebras and Leibniz algebras [165, 63].”
Trees, free right-symmetric algebras, free Novikov algebras and identities (2002, background): “The basis for the free right-symmetric algebra in terms of trees was established independently in [4].”