[Paper Review] Right Hom-alternative algebras

Donald Yau|arXiv (Cornell University)|Oct 17, 2010

Advanced Topics in Algebra14 references5 citations

TL;DR

This paper establishes foundational properties of multiplicative right Hom-alternative algebras, proving they are Hom-power associative and Hom-Jordan admissible without requiring left Hom-alternativity. It introduces Hom-versions of Albert decompositions and Moufang identities, and constructs infinite families of non-isomorphic right Hom-alternative algebras that are neither left Hom-alternative nor right alternative.

ABSTRACT

It is shown that every multiplicative right Hom-alternative algebra is both Hom-power associative and Hom-Jordan admissible. Multiplicative right Hom-alternative algebras admit Albert-type decompositions with respect to idempotents. Multiplication operators defined by idempotents in right Hom-alternative algebras are studied. Hom-versions of some well-known identities in right alternative algebras are proved.

Motivation & Objective

• To investigate structural properties of multiplicative right Hom-alternative algebras, extending classical results from alternative algebras to the Hom-algebra setting.
• To demonstrate that multiplicative right Hom-alternative algebras are Hom-power associative and Hom-Jordan admissible, even without assuming left Hom-alternativity.
• To generalize Albert’s decomposition theorem to the Hom-algebra context using idempotents.
• To establish Hom-versions of classical identities such as the Moufang identity and the Hom-Teichmüller identity in right Hom-alternative algebras.
• To construct new examples of multiplicative right Hom-alternative algebras that are neither left Hom-alternative nor right alternative, showing the class is strictly larger than previously known classes.

Proposed method

• Uses the Hom-associator definition $\text{as}(x,y,z) = (xy)\alpha(z) - \alpha(x)(yz)$ to define right Hom-alternative algebras via the identity $\text{as}(x,y,y) = 0$.
• Applies the twisting construction via self-weak morphisms to generate new right Hom-alternative algebras from existing ones, generalizing known construction results.
• Employs linearization techniques and algebraic identities, including the Hom-Teichmüller identity $\text{as}(x,y,z) + \text{as}(y,z,x) + \text{as}(z,x,y) = 0$, to derive new identities.
• Introduces the concept of a Hom-idempotent $e$ satisfying $e^2 = \alpha(e)$ and uses it to define decomposition theorems analogous to Albert’s decomposition in alternative algebras.
• Proves Hom-Moufang identities by manipulating the Hom-associator and applying the right Hom-alternative identity $\text{as}(x,y,y) = 0$ in combination with linearization.
• Constructs an infinite family of non-isomorphic multiplicative right Hom-alternative algebras via a parametric construction, showing they are not left Hom-alternative or right alternative.

Experimental results

Research questions

• RQ1 Are multiplicative right Hom-alternative algebras Hom-power associative, and can this be proven directly without relying on left Hom-alternativity?
• RQ2 Is every multiplicative right Hom-alternative algebra Hom-Jordan admissible, and does this hold independently of left Hom-alternativity?
• RQ3 Can Albert’s decomposition theorem be generalized to the Hom-algebra setting using Hom-idempotents?
• RQ4 What Hom-versions of classical identities like the Moufang identity hold in right Hom-alternative algebras?
• RQ5 Do there exist multiplicative right Hom-alternative algebras that are neither left Hom-alternative nor right alternative, and how can they be constructed?

Key findings

• Every multiplicative right Hom-alternative algebra is Hom-power associative, as proven via a short, direct argument in Theorem 3.2.
• Every multiplicative right Hom-alternative algebra is Hom-Jordan admissible, even without assuming left Hom-alternativity, as shown in Theorem 4.3.
• A generalization of Albert’s decomposition holds for multiplicative right Hom-alternative algebras, where the algebra decomposes into components related to a Hom-idempotent, as established in Proposition 5.5.
• Hom-versions of the Moufang identities and the Hom-Teichmüller identity are valid in right Hom-alternative algebras, as demonstrated through systematic algebraic manipulation.
• An infinite family of pairwise non-isomorphic multiplicative right Hom-alternative algebras is constructed, each of which is neither left Hom-alternative nor right alternative, as shown in Example 2.9.
• Multiplication operators induced by Hom-idempotents in right Hom-alternative algebras exhibit structural properties analogous to those in classical alternative algebras, as studied in Section 6.