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arXiv · 2501.05850

Partially Alternative Algebras

Tianran Hua·Ekaterina Napedenina·Marina Tvalavadze
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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.

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Tianran Hua, Ekaterina Napedenina, Marina Tvalavadze. 2025-01-10. Partially Alternative Algebras. https://doi.org/10.46298/cm.15067

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