Partially Alternative Algebras

TL;DR

This paper introduces the concept of partially alternative algebras, explores their existence in even dimensions, classifies certain algebra types, and reveals their connection to real Lie algebras, expanding algebraic theory.

Contribution

It presents the novel notion of partially alternative algebras, classifies specific cases, and establishes a link to real Lie algebras, broadening the understanding of algebraic structures.

Findings

Partially alternative algebras exist in any even dimension.

Four-dimensional partially alternative real division algebras can be simplified with an appropriate basis.

Such algebras naturally induce real Lie algebras.

Abstract

In this paper, we introduce a novel generalization of the classical property of algebras known as "being alternative," which we term "partially alternative." This new concept broadens the scope of alternative algebras, offering a fresh perspective on their structural properties. We showed that partially alternative algebras exist in any even dimension. Then we classified middle $\mathbb C$-associative (noncommutative) algebras satisfying partial alternativity condition. We demonstrated that for any four-dimensional partially alternative real division algebra, one can select a basis that significantly simplifies its multiplication table. Furthermore, we established that every four-dimensional partially alternative real division algebra naturally gives rise to a real Lie algebra, thereby bridging these two important algebraic frameworks. Our work culminates in a description of all Lie algebras arising from such partially alternative algebras. These results extend our understanding of algebraic structures and reveal new connections between different types of algebras.