Algebra and Applications 2

There also exists a twisted version of dendriform algebras, encompassing operators like the Jackson integral Iq (Ebrahimi-Fard and Manchon 2011). Returning to ordinary dendriform algebras, we observe that:

[1.124]

This identity generalizes to any number of elements, expressing the symmetrization of

in terms of the associative product and the left pre-Lie product (Ebrahimi-Fard et al . 2008). For more on dendriform algebras and the associated pre-Lie structures, see Ebrahimi-Fard et al . (2008), Ebrahimi-Fard and Manchon (2009a, 2009b, 2011) and Ebrahimi-Fard’s note in the present volume.

Post-Lie algebras have been introduced by Vallette (2007), independent of the introduction of the closely related notion of D-algebra in Munthe-Kaas and Wright (2008). A left post-Lie algebra on a field k is a k -vector space A together with a bilinear binary product ⊳ and a Lie bracket [–, –], such that

for any a , b , c ∈ A . In particular, a post-Lie algebra A is a pre-Lie algebra if and only if the Lie bracket vanishes. The space of vector fields on a Lie group is a post-Lie algebra, and the free post-Lie algebra with one generator is the free Lie algebra on the linear span of planar rooted trees. The binary product ⊳ is given by left grafting , which is not pre-Lie anymore because of planarity. This is the starting point to the Lie-Butcher theory (see, for example, Lundervold and Munthe-Kaas (2013); Ebrahimi-Fard et al . (2015); Curry et al . (2019, 2020)).

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