45 problems

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

The conjecture. Every finite-dimensional composition $\mathcal{Z}_2$-algebra is isometrically isomorphic to one of the following ten $\mathcal{Z}_2$-algebras:

A multilinear polynomial on an algebra is a polynomial whose variables occur exactly once, and its multilinear image is the set of all values obtained by evaluating it on the algeb…

Speciality conjecture. Every transposed Poisson algebra is special.

A $4$-algebra is a vector space $A$ over a field $k$ of characteristic different from $2$, equipped with a commutative bilinear multiplication satisfying … It is Guzzo's solvabilit…

Let $A$ be an algebra over a field $\mathbf{k}$ equipped with symmetric $\mathbf{k}$-linear operators $\Psi_\ell$ for $\ell=1,\ldots,m$. A Yagzhev algebra is the algebraic structur…

Let $\frak{G}$ be the Griess algebra. An associative subalgebra of $\frak{G}$ is a subalgebra whose multiplication is associative. Meyer and Neutsch established the existence of a…

Let $T$ be a simple Lie triple system and let $U(T)$ be its universal enveloping algebra. The augmentation ideal conjecture. The only proper ideal of $U(T)$ is its augmentation ide…

Let $V$ be a simple Lie triple system and let $U(V)$ be its universal enveloping algebra. The augmentation-ideal conjecture. The only proper ideal of $U(V)$ is its augmentation ide…

Let $\mathcal{E}$ be a complex evolution algebra. An element $u\in\mathcal{E}$ is an idempotent if $u^2=u$. If $\mathcal{E}$ has natural basis $\{e_1,\dots,e_n\}$ and structure mat…

Sparse-stencil conjecture. In models with sparse bracket stencils, the true convergence radius satisfies

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 the ambient rings be nonassociative rings arising in the setting of skew polycyclic codes, and let $G_{\tau,\alpha}$ denote the equivalences between such ambient rings. Isometr…

Let $\mathcal A$ be a four-dimensional real division algebra, and suppose its automorphism group is non-trivial. A reflection is an automorphism of the type considered in the paper…

Positive-signature variant conjecture. The split algebra ${\mathbb A}_{3,1}$ involves variants of ${\mathbb P}_{4}$ with positive signature.

Identity-generation conjecture. Every multilinear identity in the mutation of a $\mathfrak{perm}$ algebra over a field of characteristic zero is a consequence of

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…

Let $(D,\delta,d)$ be a nonassociative differential extension as above, with $C$ the distinguished subfield of $D$ and centralizer taken in $(D,\delta,d)$. Centralizer conjecture.…

Ternary multiplication nilpotence conjecture. The associative algebras $L(A)$ and $R(A)$ are nilpotent; in both cases, the nilpotence is of index $13$ and is not less in general.

7-Engel conjecture. The identity $t_4=0$ implies the $7$-Engel identity.

Symmetric weak nilpotence conjecture. The identity $t_4=0$ does not imply $t_5=0$, and hence does not imply the index-$4$ weak nil property; the identities $t_4=t_5=0$ imply the we…

Gröbner-basis finiteness conjecture. For each $d\in\mathbb{N}$, the reduced Gröbner basis for the nonsymmetric operad encoding the identity $w_d=0$ is finite.

Nilpotence-index-15 conjecture. The associative algebras $L(A)$ and $R(A)$ are nilpotent; in both cases, the nilpotence is of index $15$ and is not less in general.

Binary weak nilpotence conjecture. For each $n$, the identities $w_n=w_{n+1}=\cdots=w_{2n-3}=0$ imply the identity $w_{2n-2}=0$, and hence imply weak nilpotence of index $n$.