2011
DOI: 10.1090/s0002-9939-2010-10813-4
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The symmetric operation in a free pre-Lie algebra is magmatic

Abstract: Abstract. A pre-Lie product is a binary operation whose associator is symmetric in the last two variables. As a consequence its antisymmetrization is a Lie bracket. In this paper we study the symmetrization of the pre-Lie product. We show that it does not satisfy any other universal relation than commutativity. This means that the map from the free commutative-magmatic algebra to the free pre-Lie algebra induced by the symmetrization of the pre-Lie product is injective. This result is in contrast with the asso… Show more

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Cited by 14 publications
(24 citation statements)
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“…. , 20, namely 2,1,2,2,4,5,10,14,27,43,82,140,269,486,939,1765,3446,6652 with the dimensions of L n P (A, B) displayed in Table 5. They coincide up to n = 8, and obey dim L n P (A, B) < dim T n for n > 8.…”
Section: Proposition 3 Consider All Polynomial Potentialsmentioning
confidence: 99%
See 1 more Smart Citation
“…. , 20, namely 2,1,2,2,4,5,10,14,27,43,82,140,269,486,939,1765,3446,6652 with the dimensions of L n P (A, B) displayed in Table 5. They coincide up to n = 8, and obey dim L n P (A, B) < dim T n for n > 8.…”
Section: Proposition 3 Consider All Polynomial Potentialsmentioning
confidence: 99%
“…Hamiltonian) as X, but can be integrated exactly. The integrator is a composition of the form s i=1 exp(a i τ A) exp(b i τ B) = exp(Z) (1) where ∆t is the time step and exp(tX) is the time-t flow of X. The Baker-Campbell-Hausdorff formula gives Z ∈ L(A, B), the free Lie algebra with two generators.…”
Section: Introductionmentioning
confidence: 99%
“…Each (arity 3) relation in Definition 3.4 has six permutations. Let R be the matrix whose rows are the coefficient vectors of these 18 relations with respect to the ordered basis (5).…”
Section: Triassociative and Tridendriform Algebrasmentioning
confidence: 99%
“…The row space of R is the S 3 -submodule of BW(3) generated by the commutative tridendriform relations. For 1 ≤ i ≤ 27 let σ i ∈ S 3 be the permutation of the arguments in the i-th monomial (5). Let D be the diagonal matrix whose…”
Section: Triassociative and Tridendriform Algebrasmentioning
confidence: 99%
“…, X n as arguments (belonging to actual graded vector spaces on which operads act). Thus, the pre-Lie identity in the operadic form is (2) (a 1 · a 2 ) · a 3 − a 1 · (a 2 · a 3 ) = (a 1 · a 3 ) · a 2 − a 1 · (a 3 · a 2 ), and the signs in (1) arise from applying this to a decomposable tensor X 1 ⊗ X 2 ⊗ X 3 and using the standard Koszul sign rule.…”
Section: Introductionmentioning
confidence: 99%