103 problems

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 $G$ be a torsion group and let $\mathbf{F}G$ be its group algebra over a field $\mathbf{F}$. Suppose that the unit group $\mathcal{U}(\mathbf{F}G)$ satisfies a group identity.…

Let $m\in\mathbb{N}$ and let $x$ be a positive integer. The coefficients ${\mathbf{A}}_{m,0}, {\mathbf{A}}_{m,1},\ldots,{\mathbf{A}}_{m,m}$ are constants independent of $x$ and $k$…

Bahturin–Regev's conjecture. The regular quantum commutative decomposition of $R$ is minimal if and only if

Let $A$ be a PI-algebra over a field of characteristic zero, and let $c_n(A)$ denote the dimension of the space of multilinear polynomials of degree $n$ modulo the $T$-ideal of pol…

Let $F$ be a field of characteristic $0$, let $H$ be a Hopf algebra over $F$, and let $A$ be a finite-dimensional associative $H$-module algebra, with codimensions … where $P_n^H$…

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 $A$ be a $W$-algebra over a field of characteristic zero. A class of algebras has the Specht property if every $T$-ideal of the class is finitely generated. The Specht conjectu…

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…

Let $T_m(K)$ denote the algebra of upper triangular $m\times m$ matrices over a field $K$, and let $p$ be a multilinear polynomial in noncommutative variables. Fagundes–de Mello co…

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…

Let $R$ be a ring with involution, and let $R^+$ and $R^-$ denote its symmetric and skew-symmetric elements, respectively. A subset of a ring satisfies a polynomial identity if the…

Etingof–Kim–Ma conjecture.

Graded Amitsur conjecture. There exists a graded PI-exponent

Riley's conjectures. If $R$ and $S$ satisfy nonmatrix identities, then $\gamma^2(R)$ is nil of bounded index, and the tensor product

Let $\mathfrak{gl}(p,q)$ be the general linear Lie superalgebra with $p q\neq 0$, and let $\mathfrak{osp}(1,2n)$ be the orthosymplectic Lie superalgebra. A classical simple Lie sup…

Let $\Lambda=(\lambda_1,\dots,\lambda_d)$ be a finite sequence with sum … Define the polynomial $F_\Lambda(x)$ by $F_\emptyset(x)=1$ and, for $1\leq d=\sharp(\Lambda)<\infty$, … Fo…

Let $p=0$ and $n>2$. Consider the polynomial algebra $K[x_1,\ldots,x_n]$ with the Jacobian $n$-ary operation $Jac_n^S$, and, in the Poisson case, with its commutative product…

Proper containment conjecture. The ideal of polynomial identities of the path algebra is properly contained in that of its principal subalgebra:

Triple-coincidence conjecture. For every integer $n\geq4$, no three distinct pieces in $\mathcal{P}'$ share the same strength on the $n\times n$ board.

Uniqueness conjecture. The bishop–king proportionality

For each positive integer $n$, let $p_n(w)$ be the polynomial counting the LEGO structures made from $n$ parallel $w\times1$ tiles. Reflection-symmetry conjecture. For every…

Let $A$ and $B$ be … (aoverline{otimes} b)(coverline{otimes} d)=(-1)^{deg{mathbb{Z}{2}}(b)deg{mathbb{Z}{2}}(c)}(acoverline{otimes}bd). … -graded algebras; the source gives no resol…

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.