97 problems

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 $D$ be a noncommutative division algebra. A free subsemigroup is a multiplicative subsemigroup generated freely by at least two elements. Makar-Limanov's conjecture. The divisi…

Kuzmin's conjecture. The nilpotency degree satisfies

Loewy-length conjecture. If $A\sim_J B$, then

The LFED Conjecture. Images of locally finite $K$-derivations and $K$-$\mathcal{E}$-derivations of $K$-algebras are Mathieu subspaces.

Let $n$ be a fixed finite dimension, and consider the moduli space of complex Lie or associative algebras of dimension $n$. A jump deformation is a deformation that is equivalent t…

Let $\Lambda$ be a monomial triangular algebra, meaning a triangular algebra whose defining relations are paths. Its Hochschild cohomology is denoted by…

Let $\mathbb{K}$ be the ground field and let $A=\mathbb{K}\langle x_1,x_2,x_3\rangle$ be the free associative algebra on generators $x_1,x_2,x_3$. Define two operations on pairs of…

Eholzer's conjecture. The Eholzer product is associative:

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…

Let $\frak{G}$ be the Griess algebra. An associative subalgebra of $\frak{G}$ is a subalgebra whose multiplication is associative. Meyer and Neutsch established the existence of a…

Etingof–Kim–Ma conjecture.

Graded Amitsur conjecture. There exists a graded PI-exponent

Let $n\geq 1$, and consider the polynomial algebra in $n$ variables over $\mathbb C$. For suitable open subsets of $\mathbb R^n$ and positive measures, the subspace of polynomials…

Let $\mathfrak A$ be an associative algebra with a trace, and let $D_1,\dots,D_{2n}$ be derivations satisfying conditions (i)--(iii) from Section 2.4. Let $\Sigma_{N,2n-1}$ denote…

Let $\mathfrak A$ be an associative algebra with a trace, and let $D_1,\dots,D_i$ be trace-preserving associative derivations satisfying conditions (i)--(iii): each $D_j$ is trace-…

Let $V$ be a rational vertex operator algebra, and let $W_{1},\dots,W_{r}$ be a complete set of representatives of equivalence classes of irreducible $V$-modules. Here $U(V)$ is th…

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

Let $A$ be an associative algebra, let $\overline{\varphi}$ be the map in the preceding kernel-presentation conjecture, and define $\theta_2(A)=[A,A]/[A,[A,A]]$. Replacing $A$ by a…

Let $A$ be a finite-dimensional algebra over a field $\k$, let $H^*(A,A)$ denote its Hochschild cohomology ring, and let $\mathcal{N}$ be the ideal generated by the homogeneous nil…

Tensor-factorization conjecture. This map is an isomorphism of $\mathbb K$-vector spaces.

Let ${\mathcal A{\it ss}}$ be the associative operad, and let ${\mathcal B}_{{\mathcal A{\it ss}}}$ be the operad of natural operations on the cohomology of associative algebras. L…

Realization conjecture. There exists a tower of algebras $A_n$, with a basis $(e_\sigma)_{\sigma\in\mathfrak S_n}$, such that the restriction of $\rho_{m,n}$ to canonical words is…

Let $(A_n)_{n\geq 0}$ be a tower of algebras, and let $C^{(n)}$ denote the prescribed matrices associated with the tower's Cartan invariants. Existence conjecture. There exists a t…

Let $\mathfrak A$ and $\mathfrak B$ be type $1$ associative algebras whose resonant spectral values are precise in regular functionals. For an $\alpha$-regular functional, let…