56 problems

Let $F$ be an infinite field, let $n\ge 2$, and let $p\in F\langle \mathcal{X} \rangle$ be a multilinear polynomial … where $m$ is a positive integer and each $\lambda_\sigma\in F$…

Let $\mathbb{F}$ be a field, and let $l(M_n(\mathbb{F}))$ denote the length of the associative algebra of $n\times n$ matrices over $\mathbb{F}$. Paz's length conjecture. … This co…

Let $F$ be a field, let $n\ge 2$, and let $p\in F\langle x_1,\ldots,x_m\rangle$ be a multilinear polynomial. The image $p(\operatorname{M}_n(F))$ is the set of values of $p$ on the…

Let $n\geq 2$ be an integer, let $K$ be a field, and let $K\langle X\rangle$ be the free associative algebra over $K$ on a countable set of noncommutative variables. A multilinear…

Let $\mathbb K$ be a field, let $n$ be a positive integer, and let $f$ be a multilinear polynomial evaluated on the full matrix algebra $M_n(\mathbb K)$. Kaplansky–L'vov conjecture…

Let $M_n(K)$ be the algebra of $n\times n$ matrices over a field $K$ of characteristic $0$, and let $\operatorname{mindeg}(M_n(K))$ denote the minimal degree of a central polynomia…

Let $n$ be a positive integer, let $\mathbb{M}_n$ be the algebra of $n\times n$ complex matrices, and let $P\in {\mathcal{P}}(\mathbb{M}_n)$ be a projection of rank $m$, where…

Let $p$ be a multilinear polynomial in $m$ variables over a field $k$, and let $\Phi_p$ be the associated polynomial map on $M_n(k)$. Write $\mathfrak{sl}_n(k)$ for the set of trac…

Valenti–Zaicev conjecture. Every grading on $U$ is graded isomorphic to

Circulant biunitary span-condition conjecture. Every circulant biunitary satisfies the span condition, and thus is isolated among all biunitary matrices.

Let $n>5$ be prime, and let a normalized biunitary be a biunitary matrix in the chosen normalization. Popa's finiteness conjecture. For every prime $n>5$, there exist only finitely…

Let $n,m\geq2$ and let $\mathcal{H}\subset\mathbb{M}_{n,m}$ be a subspace of dimension $r$ that is isometric to a Hilbert space. The preceding theorem gives $r\leq n+m-2$. Hilbert-…

Let $K$ be a field of characteristic $0$, let $n>2$, and let $M_n(K)$ be the algebra of $n\times n$ matrices. Write $s_{2n}$ for the standard polynomial of degree $2n$. Degree-…

Nonexistence conjecture. No nontrivial $G$-grading $\Gamma$ on $M_n(K)$ satisfies the primeness property for graded central polynomials.

Let $A$ be the noncommutative reversible $4 \times 4$ matrix algebra considered in the example, and let unitary equivalence mean equivalence under conjugation by a unitary matrix.…

Scalarity conjecture. Based on the authors' investigations, one has

Let $M_n$ be the algebra of all $n\times n$ matrices over a field $\mathbb{F}$, and let … be the algebra of invariants of an $m$-tuple under algebra automorphisms. Artin's conjectu…

Rank-$k$ operator Schmidt preserver conjecture. The following conditions are equivalent:

Let $K$ be a field, let $n\geq 2$ be an integer, and let $p$ be a multilinear polynomial in noncommuting variables. Let $T_n(K)$ denote the algebra of all $n\times n$ upper triangu…

Let $S=\{A_1,\ldots,A_g\}\subseteq M_n(\mathbb{C})$ be a randomly chosen generating system, and let $\operatorname{Wie}\ell(S)$ be the least length at which words in the generators…

Let $S$ be a finite generating system of $M_n(\mathbb{C})$, with $\ell(S)$ its ordinary length and $\operatorname{Wie}\ell(S)$ its Wie-length. Linear-factor conjecture for Wie-leng…

Let $S$ be a finite subset of $M_n(\mathbb{C})$. It is a Wie-generating system if there is a $k\in\mathbb{N}$ such that words of exactly length $k$ in elements of $S$ span…

Let $K$ be a field, let $n\geq 2$ be an integer, and let $f(x_1,\dots,x_m)$ be a multilinear polynomial. Say that $f$ satisfies $(S1)$ when it is a central polynomial, so its image…

Let $K$ be a field and let $n$ be a prime number. Let $K\langle X\mid \mathbb{Z}_n\rangle$ denote the free $\mathbb{Z}_n$-graded algebra, and let…

Let $K$ be a field, let $n$ be a positive integer, and let $\mathcal{L}\subseteq \mathrm{M}_{n}(K)$ be an $\omega$-Lie-nilpotent sub Lie $K$-algebra. Dimension-bound conjecture. On…