[Paper Review] Bimodules over Hom-Jordan and Hom-alternative algebras

Sylvain Attan|arXiv (Cornell University)|Apr 3, 2018

Advanced Topics in Algebra12 references3 citations

TL;DR

This paper introduces bimodules over Hom-Jordan and Hom-alternative algebras, establishing that bimodules over classical Jordan and alternative algebras can be twisted into bimodules over their Hom-generalizations via algebra endomorphisms. The key contribution is proving that the split null extension of a Hom-Jordan algebra with its bimodule yields another Hom-Jordan algebra, generalizing classical module extensions to the Hom-algebra setting.

ABSTRACT

In this paper, bimodules over Hom-Jordan algebras and the ones over Hom-alternative algebras are defined. It is shown that bimodules over Jordan and alternative algebras are twisted into bimodules over Hom-Jordan and Hom-alternative algebras via endomorphisms respectively. Some relations between bimodules over Hom-associative, Hom-Jordan and Hom-alternative algebras, are given.

Motivation & Objective

• To define and study bimodules over Hom-alternative and Hom-Jordan algebras, extending classical module theory to the Hom-algebra framework.
• To establish a twisting construction that transforms bimodules over classical alternative and Jordan algebras into bimodules over their Hom-generalizations via endomorphisms.
• To prove that the split null extension of a Hom-Jordan algebra with a bimodule yields a new Hom-Jordan algebra, generalizing classical algebraic extensions.
• To clarify structural relationships between bimodules over Hom-associative, Hom-alternative, and Hom-Jordan algebras.

Proposed method

• Define bimodules over Hom-alternative and Hom-Jordan algebras using twisted actions that commute with the twisting map α.
• Use the anti-commutator construction x∗y = xy + yx to show that the plus algebra of a Hom-alternative algebra is a Hom-Jordan algebra.
• Prove that if a module over a classical alternative algebra is compatible with an endomorphism α, then the twisted module structure defines a bimodule over the corresponding Hom-alternative algebra.
• Establish the Hom-Jordan identity in the extended algebra A⊕V by verifying the associator vanishes using bilinearity and the compatibility of module maps with α.
• Construct the split null extension μ̃(a+m,b+n) = ab + a·n + m·b and α̃(a+m) = α_A(a) + α_V(m), showing multiplicativity and closure under the Hom-Jordan identity.
• Use the fact that α_V^n = Id_V to define iterated twisted actions ρ̃_l^n and ρ̃_r^n, proving they endow the twisted module with a bimodule structure.

Experimental results

Research questions

• RQ1 How can bimodules over classical Jordan and alternative algebras be generalized to the Hom-algebra setting?
• RQ2 Under what conditions does an endomorphism α of a classical algebra induce a bimodule structure over the corresponding Hom-algebra?
• RQ3 Can the split null extension of a Hom-Jordan algebra with a bimodule be shown to satisfy the Hom-Jordan identity?
• RQ4 What conditions ensure that a left and right special module over a Hom-Jordan algebra admits a bimodule structure?
• RQ5 How do bimodules over Hom-associative algebras relate to those over Hom-Jordan algebras?

Key findings

• A bimodule over a classical alternative algebra induces a bimodule over the corresponding Hom-alternative algebra via an endomorphism α, provided the module maps commute with α.
• The split null extension of a Hom-Jordan algebra A with a bimodule V yields a new Hom-Jordan algebra structure on A⊕V, as verified by the vanishing of the Hom-Jordan associator.
• If α_V^n = Id_V for some n ∈ ℕ, then the twisted actions ρ̃_l^n and ρ̃_r^n define a bimodule structure on the Hom-module V_α_V over the Hom-Jordan algebra A_α_A.
• A left and right special module over a Hom-Jordan algebra becomes a bimodule if it satisfies an additional compatibility condition with the twisting map α.
• The plus algebra construction (x∗y = xy + yx) turns any Hom-alternative algebra into a Hom-Jordan algebra, and bimodules over the original algebra lift to bimodules over the plus algebra.
• The Hom-associator in the extended algebra A⊕V vanishes due to the vanishing of component associators in A and the module compatibility conditions, confirming the Hom-Jordan identity holds.