24 problems

Let $F$ be a field of zero characteristic, and let $J$ be a simple Jordan superalgebra over $F$ with nonzero odd part. The possible superalgebras include $H(A,*)$, where $(A,*)$ is…

Poisson $n$-Lie algebra construction conjecture. Then $(\mathcal A, \cdot, [-,\dots,-])$ forms a Poisson $n$-Lie algebra.

Fano hyper-volumes are geometric structures associated with Cayley–Dickson algebras; in the cases discussed, the Fano plane visualises octonion structure and a 15-dimensional Fano…

Let $F$ be a field of zero characteristic, and let $J$ be a simple Jordan superalgebra over $F$ with nonzero odd part. A Jordan superalgebra is $i$-special if it is a homomorphic i…

Let $S$ be a ring with automorphism $\sigma$, let $f(t)$ be the polynomial defining the source quotient, and let $a,b\truein S$ and $\mathbf{m},\mathbf{n}$ be the associated parame…

Commutative loop conjecture. The structure $(3^{A},\oplus)$ satisfies:

Let $\mathbb{K}$ be the underlying field. The non-commutative group spherical spaces described in the source include non-degenerate quaternion algebras on $\mathbb{K}^4$ and non-tr…

Let $V_1$ and $V_2$ be vector spaces, and let $b:V_1\times V_2\to\mathbb{K}$ be a bilinear form of rank $2$ that is degenerate, so that $V_1$ or $V_2$ has dimension greater than…

Let $\mathcal{L}=(\{1,a,b,c,d\},\ast)$ be the smallest loop that is not a group, with multiplication operation $\ast$. Loop-switch strategy conjecture. There exists a nontrivial ad…

A gyrogroup is a set equipped with a binary operation and the associated gyroautomorphisms satisfying the gyrogroup axioms; it is commutative when its binary operation satisfies…

Let $L$ be an elliptic loop over $\mathbb{Z}/p^e\mathbb{Z}$. The maximality conjecture for elliptic loops. Layers and groups of size $p^{e-1}$ at infinity are the largest possible…

Octonion image conjecture. The image of this evaluation is a real vector subspace of $\mathbb{O}$.

Generalized Kaplansky conjecture. The evaluation of any multilinear polynomial over $A$ is a vector subspace of $A$.

Cyclic-generated submodule conjecture. The submodule generated by $m$ satisfies

Let $Q(*)$ be a quasigroup, meaning that for all $a,b\in Q$ there exist unique $x,y\in Q$ such that $a*x=b$ and $y*a=b$. A triple $(x,y,z)\in Q^3$ is associative if…

Let $A=Ke\oplus U\oplus V$ be a finitely generated Bernstein algebra that is nuclear, and let $N=U\oplus V$ be its barideal. Grishkov's conjecture. The barideal $N$ is nilpotent. T…

Doro's conjecture. If $\operatorname{Nuc}(M)=1$, then $\operatorname{C}(M)$ is normal in $M$.

Minimal dimension conjecture. Every Malcev algebra of the second type of dimension $<23$ over a field of characteristic $0$ is an algebra of the first type.

Group-table modification conjecture. There exists a group $G$ and a fixed central involution $h\in G$ such that $C=(G,*)$ and, for all $x,y\in G$,

The Main Conjecture. The quotient $Q/N(Q)$ is an abelian group and $Q/Z(Q)$ is a group. In particular, $Q$ is nilpotent of class at most $3$.

Finite-generation conjecture. The known identities of degree $\leq 6$ generate all polynomial identities of $\mathbb{O}$.

Solvable Sabinin algebra conjecture. There should be a version of Theorem for solvable Sabinin algebras.

Let $A_{p,q}$ ($p \le q$) be the simple associative triple system consisting of $(p+q) \times (p+q)$ block matrices of the form … Let $\omega$ be one of the following trilinear ope…

Let $u\in {\mathcal{C}}\ell_{0,7}$ and let $A,B\in\mathbb{O}$, where $\mathbb{O}$ is the octonion algebra. Write $\bullet_\llcorner$ and $\bullet_\lrcorner$ for the corresponding l…