19 problems

Frobenius-form conjecture. Every axial algebra of Monster type admits a Frobenius form.

Griess-algebra dimension conjecture. In this case, the Griess algebra has the largest possible dimension among three-generated algebras of Monster type, excluding Matsuo algebras a…

Let $A$ be an algebra of Jordan type $\frac{1}{2}$. The original conjecture was that $A$ is either a Matsuo algebra, or a factor of a Matsuo algebra, or a Jordan algebra. Desmet's…

Let $A$ be an algebra of Jordan type with four generating axes, and suppose its universal algebra has a principal component, meaning a multidimensional component from which all or…

Kernel conjecture. The kernel $K$ coincides with $\hat P$.

Connectedness conjecture. There are generic examples which are connected in this axial graph.

Franchi–Mainardis–Shpectorov conjecture. Every axial algebra of monster type has a Frobenius form.

Let $A$ be a $\mathcal{J}(\alpha,\beta)$-algebra generated by four primitive axes. Here, $\mathcal{J}(\alpha,\beta)$ denotes the class of algebras under consideration with eigenval…

Projective-group conjecture. There exist parameters $\alpha,\beta,\gamma,\psi\in F$ such that $T\simeq PSL(3,F)$, and a different set of parameters such that $T\simeq PSU(3,F)$. Th…

Let $a$ and $b$ be axes in a pseudo-composition algebra over $4\mathbb{R}4$, let $\tau_a$ and $\tau_b$ be their Miyamoto involutions, and set $\tau_{ab}=\tau_a\tau_b$. The Miyamoto…

McInroy–Shpectorov conjecture. Every connected Jordan type $\eta = \tfrac{1}{2}$ algebra is either a Jordan algebra or a quotient of a Matsuo algebra.

Solid-subalgebra conjecture. Then $A$ is a Jordan algebra.

Classification conjecture. Every algebra of Jordan type is either a Jordan algebra, a Matsuo algebra, or a factor of a Matsuo algebra.

Let $\mathbb{F}$ be a field of characteristic not equal to two, and let $A$ be a commutative primitive axial algebra over $\mathbb{F}$ of Jordan type $\frac{1}{2}$, meaning that…

Let $F$ be a field of characteristic different from $2$. Let $J$ be a Jordan algebra over $F$ generated by idempotents, and let a $\mathcal{J}(\frac{1}{2})$-axial central extension…

Axis conjecture. By solving a suitable further system of equations, the idempotent $b$ could also be made into an axis.

Let $A$ be a $2$-generated axial algebra of Monster type over a field of characteristic $0$. Sakuma's conjecture. The algebra $A$ is one of the nine Norton–Sakuma algebras. A class…

Let $A$ be an axial algebra of Monster-type with generating axes $X$, and let $\Delta(X)$ be the graph whose vertices are the axes in $X$, with distinct vertices $a,b$ adjacent exa…

Let $A$ be a dihedral axial algebra of Monster type over a field of characteristic $0$. The nine Norton–Sakuma algebras are the nine classified dihedral algebras arising in this se…