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Partially Alternative Algebras

Tianran Hua, Ekaterina Napedenina, Marina Tvalavadze

TL;DR

This work generalizes alternativity through the notion of partially alternative algebras, develops the theory around imaginary units and the commutative nucleus, and establishes existence in even dimensions. Focusing on four‑dimensional real structures, it analyzes middle $\mathbb{C}$‑associative algebras, provides a classification into algebras such as $\mathcal{M}^+$, $\mathcal{M}^0$, and $\mathbb{H}$, and connects these algebras to Lie theory via the commutator bracket, producing a taxonomy of the associated Lie algebras. It further shows that, under division and reflection assumptions, four‑dimensional partially alternative algebras reduce (up to isomorphism) to known algebras and yield Lie algebras aligned with Mubarakzyanov's taxonomy, including direct sums with $\mathfrak{so}(3)$ and solvable cases. Overall, the paper reveals structural bridges between nonassociative algebras and classical Lie theory, expanding the landscape of alternativity generalizations and their interplays.

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.

Partially Alternative Algebras

TL;DR

This work generalizes alternativity through the notion of partially alternative algebras, develops the theory around imaginary units and the commutative nucleus, and establishes existence in even dimensions. Focusing on four‑dimensional real structures, it analyzes middle ‑associative algebras, provides a classification into algebras such as , , and , and connects these algebras to Lie theory via the commutator bracket, producing a taxonomy of the associated Lie algebras. It further shows that, under division and reflection assumptions, four‑dimensional partially alternative algebras reduce (up to isomorphism) to known algebras and yield Lie algebras aligned with Mubarakzyanov's taxonomy, including direct sums with and solvable cases. Overall, the paper reveals structural bridges between nonassociative algebras and classical Lie theory, expanding the landscape of alternativity generalizations and their interplays.

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 -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.
Paper Structure (5 sections, 14 theorems, 37 equations)

This paper contains 5 sections, 14 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Definitions and preliminary example
  3. Middle C-associative algebras
  4. Partially alternative division algebras
  5. Lie algebras associated with partially alternative algebras

Key Result

theorem 1

If $\mathcal{J}$ is a simple nearly alternative algebra with an idempotent $e\neq 1$, then $\mathcal{J}$ is commutative or alternative.

Theorems & Definitions (29)

  • theorem 1
  • definition 1
  • definition 2
  • definition 3
  • proposition 1
  • proof
  • proposition 2
  • proof
  • lemma 1
  • proof
  • ...and 19 more