[Paper Review] Hom-Lie-Yamaguti Superalgebras

Donatien Gaparayi, Sylvain Attan|arXiv (Cornell University)|Nov 1, 2017

Advanced Topics in Algebra11 references3 citations

TL;DR

This paper introduces Hom-Lie-Yamaguti superalgebras as a Hom-type generalization of Lie-Yamaguti superalgebras, unifying Hom-Lie superalgebras and Hom-Lie supertriple systems. It establishes that the category of multiplicative Hom-Lie-Yamaguti superalgebras is closed under twisting by self-morphisms and under the $n^{th}$-derived construction, extending the theory of derived algebras to binary-ternary Hom-superalgebras.

ABSTRACT

(Multiplicative) Hom-Lie-Yamaguti superalgebras which generalize Hom-Lie supertriple systems (and subsequently ternary multiplicative Hom-Nambu superalgebras) and Hom-Lie superalgebras in the same way as Lie-Yamaguti superalgebras [Frac] generalize Lie supertriple systems and Lie superalgebras are defined. We show that the category of (multiplicative) Hom-Lie-Yamaguti superalgebras is closed under twisting by self-morphisms. Construction of some examples of Hom-Lie-Yamaguti superalgebras is given. The notion of an $n^{th}-$derived (binary) Hom-superalgebras is extended to the one of an $n^{th}-$derived binary-ternary Hom-superalgebras and it is shown that the category of Hom-Lie-Yamaguti superalgebras is closed under the process of taking $n^{th}-$ derived Hom-superalgebras.

Motivation & Objective

• To define and formalize Hom-Lie-Yamaguti superalgebras as a generalization of Hom-Lie superalgebras and Hom-Lie supertriple systems.
• To extend the notion of $n^{th}$-derived (binary) Hom-superalgebras to the binary-ternary case, introducing $n^{th}$-derived binary-ternary Hom-superalgebras.
• To prove that the category of multiplicative Hom-Lie-Yamaguti superalgebras is closed under the process of taking $n^{th}$-derived algebras.
• To demonstrate that twisting by self-morphisms preserves the Hom-Lie-Yamaguti superalgebra structure.
• To show that special cases of the $n^{th}$-derived construction recover known structures: Hom-Lie supertriple systems and Hom-Lie superalgebras when the binary or ternary operation vanishes, respectively.

Proposed method

• Define a Hom-Lie-Yamaguti superalgebra as a $\mathbb{Z}_2$-graded vector space $L = L_0 \oplus L_1$ equipped with a binary superoperation $*$ and a ternary superoperation $\{, ,\}$, satisfying graded skew-symmetry, super-Jacobi identity, and generalized Leibniz-type identities.
• Introduce the twisting map $\alpha$ as a linear self-map that deforms the original identities of Lie-Yamaguti superalgebras, recovering the original structure when $\alpha = \text{id}$.
• Construct the $n^{th}$-derived Hom-superalgebra $A^{(n)}$ via iterated application of the twisting map $\alpha$, defining new operations $[x,y]^{(n)} = \alpha^{2^n - 1}([x,y])$ and $\{x,y,z\}^{(n)} = \alpha^{2^n - 1}(\{x,y,z\})$.
• Verify that the $n^{th}$-derived algebra $A^{(n)}$ satisfies all defining identities of a Hom-Lie-Yamaguti superalgebra by induction and using the multiplicativity of $\alpha$.
• Use the multiplicativity of $\alpha$ to transfer identities from the original algebra $A_\alpha$ to $A^{(n)}$, ensuring closure under the derived construction.
• Apply the supercommutator bracket and ternary operations in the derived algebra, showing that identities such as $(SHLY6)$, $(SHLY7)$, and $(SHLY8)$ are preserved through $\alpha$-twisting and iteration.

Experimental results

Research questions

• RQ1 How can the Hom-type generalization of Lie-Yamaguti superalgebras be systematically defined to unify Hom-Lie superalgebras and Hom-Lie supertriple systems?
• RQ2 Is the category of multiplicative Hom-Lie-Yamaguti superalgebras closed under the operation of twisting by self-morphisms?
• RQ3 Can the concept of $n^{th}$-derived (binary) Hom-superalgebras be extended to the binary-ternary setting, and does it preserve the Hom-Lie-Yamaguti superalgebra structure?
• RQ4 What happens to the $n^{th}$-derived algebra when the binary or ternary operation vanishes, and does it recover known structures like Hom-Lie supertriple systems or Hom-Lie superalgebras?
• RQ5 Under what conditions does the $n^{th}$-derived construction preserve the defining identities of Hom-Lie-Yamaguti superalgebras?

Key findings

• The category of multiplicative Hom-Lie-Yamaguti superalgebras is closed under twisting by self-morphisms, meaning that applying a self-morphism $\alpha$ to a Hom-Lie-Yamaguti superalgebra yields another Hom-Lie-Yamaguti superalgebra.
• The $n^{th}$-derived construction preserves the Hom-Lie-Yamaguti superalgebra structure: if $A_\alpha$ is a multiplicative Hom-Lie-Yamaguti superalgebra, then its $n^{th}$-derived algebra $A^{(n)}$ is also a Hom-Lie-Yamaguti superalgebra.
• When the binary operation $*$ vanishes in $A_\alpha$, the $n^{th}$-derived algebra becomes a Hom-Lie supertriple system, and hence also a ternary Hom-Nambu superalgebra.
• When the ternary operation $\{, ,\}$ vanishes in $A_\alpha$, the $n^{th}$-derived algebra reduces to a Hom-Lie superalgebra.
• The verification of identities such as $(SHLY6)$, $(SHLY7)$, and $(SHLY8)$ in $A^{(n)}$ relies on the multiplicativity of $\alpha$ and the transfer of identities from $A_\alpha$ via $\alpha$-twisting.
• The $n^{th}$-derived algebra $A^{(n)}$ is constructed via $[x,y]^{(n)} = \alpha^{2^n - 1}([x,y])$ and $\{x,y,z\}^{(n)} = \alpha^{2^n - 1}(\{x,y,z\})$, ensuring compatibility with the original algebraic structure.