Axial algebras are non-associative algebras generated by semisimple idempotents, known as axes, that all obey a fusion rule. Axial algebras were introduced by Hall, Rehren and Shpectorov as a generalisation of the axioms of Majorana theory, which was in turn introduced by Ivanov as an axiomatisation of certain properties of the 2A-axes of the Griess algebra. Axial algebras of Monster type are axial algebras whose axes obey the Monster, or Majorana, fusion rule. We construct an axial algebra of Monster type $M_{4A}$ over the polynomial ring $\mathbb{R}[t]$ that is generated by six axes whose Miyamoto involutions generate an elementary abelian group of order $4$. This construction automatically provides an infinite-parameter family $\{M(t)\}_{t \in \mathbb{R}}$ of axial algebras of Monster type each of which admit a unique Frobenius form. Moreover, we show that this form on $M(t)$ is positive definite if and only if $0 < t < \frac{1}{6}$ and also satisfies Norton's inequality if and only if $0 \leq t \leq \frac{1}{6}$. Finally, we show that the $4A$ axes of $M_{4A}$ obey a $C_2 \times C_2$-graded fusion rule giving a new infinite family of fusion rules.

中文翻译：

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一个无限的轴向代数族

轴代数是由半单幂等生成的非关联代数，称为轴，它们都遵循融合规则。轴代数由 Hall、Rehren 和 Shpectorov 引入，作为 Majorana 理论公理的推广，而 Ivanov 又将其引入作为 Griess 代数的 2A 轴的某些性质的公理化。Monster 类型的轴代数是轴遵循 Monster 或 Majorana 融合规则的轴代数。我们在多项式环 $\mathbb{R}[t]$ 上构建了一个 Monster 类型的轴向代数 $\mathbb{R}[t]$，该多项式环由六个轴生成，其 Miyamoto 对合生成一个 $4$ 阶的基本阿贝尔群。这种构造自动提供了一个无限参数族 $\{M(t)\}_{t \in \mathbb{R}}$ 的 Monster 类型的轴向代数，每个都承认一个独特的 Frobenius 形式。而且，我们证明 $M(t)$ 上的这种形式是正定的当且仅当 $0 < t < \frac{1}{6}$ 并且也满足诺顿不等式当且仅当 $0 \leq t \leq \frac{ 1}{6}$。最后，我们表明 $M_{4A}$ 的 $4A$ 轴遵循 $C_2 \times C_2$ 分级融合规则，给出了一个新的无限融合规则系列。