19 problems

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

Let $\mathbb{F}$ be an arbitrary field, and let $\ell(\mathrm{M}_n(\mathbb{F}))$ denote the length of the full matrix algebra $\mathrm{M}_n(\mathbb{F})$. Paz's length conjecture. ……

L'vov–Kaplansky conjecture. If $f$ is multilinear, then its image in the matrix algebra $M_n(K)$ is a vector space for every $n\in\mathbb{N}$.

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 $n\geq 2$ 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 $S$ 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 $K$ 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, let $n\geq 2$ and $m\geq 1$ be integers, and let $f(x_1,\dots,x_m)$ be a nonzero multilinear polynomial in $K\langle x_1,\dots,x_m\rangle$. Mesyan conjecture. I…

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 $p$ 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…

Equivalent L'vov–Kaplansky conjecture. The image of $p$ satisfies

Let $K$ be a field of characteristic $0$, and let $M_d(K)$ denote the algebra of $d\times d$ matrices. A central polynomial for $M_d(K)$ is a polynomial that takes central values o…

Let $K$ be an infinite field of characteristic $2$, and let $M_2(K)$ denote the algebra of $2\times 2$ matrices. A polynomial identity basis is a finite set of polynomial identitie…

Let $R$ be a commutative unital ring, let $m,n\geq 1$, and let $E^{m}$ be the $m$-generated Grassmann algebra … For $k\geq 1$, the standard polynomial is … It is a standard identit…

Common-unitary normal-form conjecture. Any algebra $\mathfrak A\subseteq\mathfrak C_D$ induced by such $\mathscr A$ and $s$ admits a unitary sequence $\{U_n\}_n$ such that…

Kippenhahn's conjecture. The algebra generated by $H$ and $K$ cannot be the full algebra $M_{2n}(\mathbb{C})$.

Kye's conjecture. Every 2-positive, respectively 2-copositive, linear map in $B(M_3(\mathbb{C}),M_3(\mathbb{C}))$ is decomposable.

Let $f$ be a multilinear polynomial with complex coefficients, and let $d\in\mathbb{N}$. Consider the set of values obtained by evaluating $f$ on $M_d(\mathbb{C})$. Kaplansky–Lvov…

Let ${\mathfrak D}$ denote the set of block-positive symmetries under consideration, and let $s\in{\mathfrak D}$ be a partial symmetry for $n=3$. Partial-symmetry conjecture. The r…