[Paper Review] Hom-Lie-Yamaguti structures on Hom-Leibniz algebras

Donatien Gaparayi, A. Nourou|arXiv (Cornell University)|Aug 29, 2012

Advanced Topics in Algebra25 references3 citations

TL;DR

This paper establishes that every multiplicative left Hom-Leibniz algebra naturally carries a Hom-Lie-Yamaguti (Hom-LY) algebra structure through a skew-symmetrization and ternary operation construction. The key result is an α-twisted generalization of the classical Lie-Yamaguti structure on Leibniz algebras, preserving the algebraic identities under the twisting map α.

ABSTRACT

Multiplicative left Hom-Leibniz algebras have natural Hom-Lie-Yamaguti structure.

Motivation & Objective

• To extend the classical Lie-Yamaguti algebra structure from Leibniz algebras to the Hom-algebra setting.
• To establish a systematic construction of Hom-Lie-Yamaguti algebras from multiplicative left Hom-Leibniz algebras.
• To generalize the known LY-algebra structure on Leibniz algebras by introducing a twisting map α.
• To provide explicit examples of Hom-LY algebras derived from known Hom-Leibniz algebras.
• To demonstrate that the Hom-LY identities are preserved under the multiplicative condition and α-twisting.

Proposed method

• Define the binary operation [x,y] := x·y - y·x as the skew-symmetrization of the Hom-Leibniz product.
• Define the ternary operation {x,y,z} := -(x·y)·z for the Hom-Lie-Yamaguti structure.
• Verify that the resulting (L, [·,·], {·,·,·}, α) satisfies all Hom-LY algebra axioms: (HLY1)–(HLY8).
• Use the multiplicativity of α (α(x·y) = α(x)·α(y)) to ensure compatibility with the twisted identities.
• Construct examples by selecting endomorphisms α of known left Leibniz algebras and defining the twisted product x·α y := α(x·y).
• Confirm that the resulting (L, ·α, α) is a multiplicative left Hom-Leibniz algebra, enabling the Hom-LY structure.

Experimental results

Research questions

• RQ1 Can a Hom-Lie-Yamaguti algebra structure be naturally induced on any multiplicative left Hom-Leibniz algebra?
• RQ2 How does the classical Lie-Yamaguti structure on Leibniz algebras generalize under the Hom-algebra twisting procedure?
• RQ3 What are the necessary and sufficient conditions on α for the induced Hom-LY structure to satisfy all defining identities?
• RQ4 How can explicit examples of Hom-LY algebras be constructed from known Hom-Leibniz algebras?
• RQ5 Is the Hom-LY structure unique and canonical for a given multiplicative left Hom-Leibniz algebra?

Key findings

• Every multiplicative left Hom-Leibniz algebra (L, ·, α) admits a canonical Hom-Lie-Yamaguti structure defined by [x,y] = x·y - y·x and {x,y,z} = -(x·y)·z.
• The resulting (L, [·,·], {·,·,·}, α) satisfies all eight Hom-LY algebra axioms (HLY1)–(HLY8), including α-twisted Jacobi and ternary identities.
• When α is the identity map, the construction recovers the classical Lie-Yamaguti structure on Leibniz algebras as in [14].
• Explicit examples of Hom-LY algebras are constructed from 3- and 4-dimensional complex left Leibniz algebras with specific endomorphisms α.
• The construction yields nontrivial Hom-LY algebras, such as one with [e₁,e₂] = 2(e₃+e₄) and {e₁,e₂,e₁} = e₄ in a 4D nilpotent algebra.
• The method provides a systematic way to generate new Hom-LY algebras from known Hom-Leibniz algebras via endomorphism-induced twisting.