53 problems

A Moufang set is a permutation structure with associated root groups; a Moufang set is proper when its associated group is not sharply $2$-transitive. For a quadratic Jordan divisi…

Let $A$ be an associative algebra over a commutative ring $R$. Define the Lie product by $[x,y]=xy-yx$ and the Jordan product by $x\circ y=xy+yx$ for $x,y\in A$; mappings preservin…

For row $\#8$, let $\mathfrak J=F\oplus Q_n$ with $n\geq 3$, and let the associated representation be $V_2\otimes V(\omega_1)$ of ${\rm SL}_2\times {\rm SO}_{n+2}$, as in the table…

Equivalence conjecture. $A$ is zJpd if and only if it is zpd.

Let $V$ be a finite-dimensional vector space, let $\operatorname{Jord}(V)$ be the free Jordan algebra generated by $V$, and let $a(V)$ and $b(V)$ be the unique elements of the augm…

Let $V$ be a finite-dimensional vector space, let $\operatorname{Jord}(V)$ be the free Jordan algebra generated by $V$, and let $\mathsf{TAG}(\operatorname{Jord}(V))$ be its functo…

Kashuba--Mathieu's conjecture. The homology $H_\bullet(\mathsf{TAG}(\operatorname{Jord}(V)),\mathbbold{k})$ has no trivial or adjoint $\mathfrak{sl}_2$-module components in homolog…

Let $J(D)$ be the free Jordan $\mathbb{K}$-algebra on $D$ generators, and let $\widehat{\mathfrak{sl}}_2(J(D))$ denote the associated Lie algebra. Write $H_k(-)$ for its homology g…

Quadratic Jordan algebra conjecture. Every proper Moufang set with abelian root groups comes from a quadratic Jordan division algebra.

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 $J(r)$ be the free Jordan algebra of rank $r$, and let $\mathcal{C}^{fin}$ be the category of finite-dimensional $\mathbb{Z}$-graded $\mathfrak{sl}_2(J(r))$-modules. For each d…

Let $J(r)$ be the free Jordan algebra of rank $r$, and let $\mathfrak{sl}_2(J(r))$ be its universal central extension of the Tits–Kantor–Koecher algebra. The Lie 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…

A JB$^*$-algebra is a Jordan–Banach $^*$-algebra. An element is nilpotent if some positive power of it is zero; it is non-trivial when it is nonzero. The nilpotent characterization…

Let $A$ be a unital Jordan algebra, and consider trigonometric and quasi-trigonometric solutions of the $A$-classical Yang–Baxter equation (A-CYBE). Jordan unitality obstruction co…

Let $L$ be a finite-dimensional Lie algebra over an algebraically closed field of characteristic not equal to $2$, and let $\operatorname{Rad}(L)$ be its solvable radical. The quot…

Let $L$ be a Lie algebra, and let the property $\heartsuit$ denote the property discussed in the source. A subalgebra has codimension $1$ when its quotient vector space has dimensi…

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…

Let $A$ be a Jordan PI-algebra, and let its codimension growth be measured by the sequence of ordinary codimensions. The scale 2 referred to in the paper is the scale for codimensi…

Nonexistence conjecture. For $n=6$, one has

Polynomial codimension conjecture. The codimensions of $A^{(2)}_{r,k_1,k_2}$, $A^{(3)}_{k_1,k_2,k_3}$, and $B^{(2)}_{k,\ell_1,\ell_2}$ in…

Exclusion of true spin-factors. No true spin-factors are allowed. Equivalently, the resulting JW-algebras should restrict to the universally reversible ones.

The exceptional Jordan algebra is the unique exceptional simple Jordan algebra, realized as the algebra of $3\times 3$ Hermitian octonionic matrices. It gives rise to the exception…