Lie n-racks

Generalizations of Laver tables

“…Definition 2. 5. If X is an LD-system and x ∈ X, then we shall say that the element x is a left-identity if x * y = y for each y ∈ X.…”

“…Therefore the ternary self-distributive algebras are motivated as being algebras with many inner endomorphisms. Ternary selfdistributive algebras (in particular, ternary racks and quandles) have been studied in the papers [22] and [5]. Furthermore, multi-LD-systems were studied in [14].…”

“…Definition 4. 5. If X = (X, (t X ) t∈E ) is an algebra where each t ∈ E has arity n t + 1, then define the hull Γ(X ) to be the algebra (( t∈E {t} × X nt , * ) where * is defined by (s, x 1 , .…”

Lie Central Triple Racks

“…Partial solutions to this problem for Leibniz algebras (dubbed by Loday as the Coquecigrue problem) have been provided by several authors (see M. Kinyon [10], S. Covez [6]). The author extended Kinyon's results to Leibniz 3-algebras using Lie 3-racks [2]. In this paper we open the problem of integration of gb-triple systems.…”

“…Let (R, [−, −, −] R , 1) be a Lie c-triple rack and g := T 1 R. Then for all a, c ∈ R, the tangent mapping φ (a,c) * = T 1 (φ (a,c) ) is an automorphism of g.Proof.Let X, Y, Z ∈ g and let x, y, z be respectively the images of X, Y and Z by the exponential map exp 1 (seeRemark 3.5). By the c-distributive property of c-triple racks, we haveφ (a,c) (φ (x,z) (y)) = φ (φ (a,c) (x),φ (a,c) (z)) (φ (a,c) (y))which when successively differentiated at 1 ∈ R with respect to the parameter γ Y then γ Z then γ X yieldsφ (a,c) * ([X, Y, Z] g ) = [φ (a,c) * (X), φ (a,c) * (Y ), φ (a,c) * (Z)] g(2). Let R be a Lie c-triple rack and A, C ∈ g := T 1 R. Let a, c be respectively the images of A and C by the exponential map exp 1 .Then the mappingD (A,C) * : g −→ gl(g) is a derivation of g. Moreover, D (A,C) * is exactly T 1 (Φ), where Φ is the mapping Φ : R × R −→ GL(g) defined by Φ(a, c) = φ (a,c) * .Proof.…”