The symmetric operation in a free pre-Lie algebra is magmatic

Bergeron, Nantel; Loday, Jean-Louis

doi:10.1090/s0002-9939-2010-10813-4

Cited by 14 publications

(24 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…. , 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

The Lie algebra of classical mechanics

McLachlan¹,

Murua²

2019

Journal of Computational Dynamics

View full text Add to dashboard Cite

Classical mechanical systems are defined by their kinetic and potential energies. They generate a Lie algebra under the canonical Poisson bracket. This Lie algebra, which is usually infinite dimensional, is useful in analyzing the system, as well as in geometric numerical integration. But because the kinetic energy is quadratic in the momenta, the Lie algebra obeys identities beyond those implied by skew symmetry and the Jacobi identity. Some Poisson brackets, or combinations of brackets, are zero for all choices of kinetic and potential energy, regardless of the dimension of the system. Therefore, we study the universal object in this setting, the 'Lie algebra of classical mechanics' modelled on the Lie algebra generated by kinetic and potential energy of a simple mechanical system with respect to the canonical Poisson bracket. We show that it is the direct sum of an abelian algebra X, spanned by 'modified' potential energies isomorphic to the free commutative nonassociative algebra with one generator, and an algebra freely generated by the kinetic energy and its Poisson bracket with X. We calculate the dimensions c n of its homogeneous subspaces and determine the value of its entropy lim n→∞ c 1/n n . It is 1.8249 . . . , a fundamental constant associated to classical mechanics. We conjecture that the class of systems with Euclidean kinetic energy metrics is already free, i.e., the only linear identities satisfied by the Lie brackets of all such systems are those satisfied by the Lie algebra of classical mechanics.

show abstract

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%

The Lie algebra of classical mechanics

McLachlan¹,

Murua²

2019

Journal of Computational Dynamics

View full text Add to dashboard Cite

show abstract

“…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%

Jordan trialgebras and post-Jordan algebras

2017

View full text Add to dashboard Cite

We compute minimal sets of generators for the S n -modules (n ≤ 4) of multilinear polynomial identities of arity n satisfied by the Jordan product and the Jordan diproduct (resp. pre-Jordan product) in every triassociative (resp. tridendriform) algebra. These identities define Jordan trialgebras and post-Jordan algebras: Jordan analogues of the Lie trialgebras and post-Lie algebras introduced by Dotsenko et al., Pei et al., Vallette & Loday. We include an extensive review of analogous structures existing in the literature, and their interrelations, in order to identify the gaps filled by our two new varieties of algebras. We use computer algebra (linear algebra over finite fields, representation theory of symmetric groups), to verify in both cases that every polynomial identity of arity ≤ 6 is a consequence of those of arity ≤ 4. We conjecture that in both cases the next independent identities have arity 8, imitating the Glennie identities for Jordan algebras. We formulate our results as a commutative square of operad morphisms, which leads to the conjecture that the squares in a much more general class are also commutative. ASSOCIATIVE ALGEBRASSelf-dual Operation ab Relation (ab)c ≡ a(bc) Lie bracket [a, b] = ab − ba symmetry ab ≡ ba, relation (ab)c ≡ a(bc) Lie algebras, equation (1) Commutative associative algebras Jordan product a • b = ab + ba Jordan algebras, equation (2) No dual, operad is cubic not quadratic DIASSOCIATIVE ALGEBRAS DENDRIFORM ALGEBRAS Operations a b, a b, Definition 2.9 Operations a ≺ b, a b, Definition 2.9 Leibniz bracket {a, b} = a b − b a Leibniz algebras, Definition 2.12 Zinbiel algebras, Definition 2.12 Pre-Lie product {a, b} = a ≺ b − b a Perm algebras, Definition 2.16 Pre-Lie algebras, Definition 2.16 Jordan diproduct a • b = a b + b a Jordan dialgebras, Definition 2.18 No dual, operad is cubic not quadratic Pre-Jordan product a • b = a ≺ b + b a No dual, operad is cubic not quadratic Pre-Jordan algebras, Definition 2.18 TRIASSOCIATIVE ALGEBRAS TRIDENDRIFORM ALGEBRAS Operations a b, a ⊥ b, a b Operations a ≺ b, a b, a b Definition 3.1 Definition 3.1 Lie bracket [a, b] = a ⊥ b − b ⊥ a Leibniz bracket {a, b} = a b − b a Commutative tridendriform algebras, Lie trialgebras, Definition 3.4 Definition 3.4 Lie bracket [a, b] = a b − b a Commutative triassociative algebras, Pre-Lie product {a, b} = a ≺ b − b a Definition 3.8 Post-Lie algebras, Definition 3.8 Jordan product a • b = a ⊥ b + b ⊥ a Jordan diproduct a • b = a b + b a Jordan trialgebras, Section 4 No dual, operad is cubic not quadratic Jordan product a • b = a b + b a Pre-Jordan product a • b = a ≺ b + b a No dual, operad is cubic not quadratic Post-Jordan algebras, Section 6

show abstract

“…, 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%