An infinite family of axial algebras

Abstract: 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.