Automorphism group of a Matsuo algebra

Abstract

Axial algebras are commutative, not necessarily associative algebras generated by axes, where axes are idempotents whose adjoint actions are semisimple and such that the product of eigenvectors is controlled by a specific fusion law. Matsuo algebras, defined using Fischer spaces and $3$-transposition groups, are examples of axial algebras of Jordan type $\eta$. In this thesis, we show that Matsuo algebras with $\eta \neq \frac{1}{2}$ almost never contain additional axes, and consequently, their automorphism groups almost always coincide with the automorphism group of the underlying groups of $3$-transpositions. Furthermore, we identify all exceptions to this general rule in the irreducible case. Namely, we find three infinite series and an additional sporadic example of Matsuo algebras having extra axes and a larger group of automorphisms. In particular, for specific $\eta$, we show that $M=M_\eta(S_n)$ is isomorphic to a quotient of $M'=M_\eta(S_{n+1})$, and similarly $M=M_\eta(O^{\epsilon}_{2m}(2))$ is isomorphic to a quotient of $M'=M_\eta(Sp_{2m}(2))$. These are examples of what we call aligned pairs of groups of $3$-transpositions. We use the information about the spectrum of the diagram of irreducible $3$-transposition groups, provided in \cite{JS}, to find all aligned pairs of irreducible $3$-transposition groups. We conclude that, apart from the examples above, there is only one additional sporadic aligned pair $(Fi_{23},\Omega_8(3):S_3)$.