45 problems

Finite basis conjecture. The algebra $M_2(K)$ does not have a finite basis of its polynomial identities.

Let $L$ be a Lie $F$-algebra acting on a finite-dimensional algebra $A$ by derivations, so that $A$ is an $L$-algebra. Let $c_n(A)$ and $c_n^L(A)$ denote the ordinary and different…

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…

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…

Smallness equivalence conjecture. A unital pseudoalgebra $\operatorname{Diff} A$ is small if and only if $A$ is a small $X^{cop}$-simple algebra.

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

Let $p=2$, let $R$ be the truncated polynomial ring from the construction, let $\mathcal L=\operatorname{Lie}(v_1,v_2)\subset\operatorname{Der}R$ be the Fibonacci Lie algebra, and…

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…

Amitsur's conjecture. The limit exists and belongs to $\mathbb Z_+$.

Let $A$ be a finitely generated PI-algebra over the field of rational numbers, and let $J$ be its Jacobson radical. Representability conjecture. If … then $A$ is representable. Thi…

Alahmadi–Alsulami–Jain–Zelmanov conjecture. The following are equivalent:

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

Isomorphism conjecture. The three algebras are isomorphic as associative algebras:

Let $\mathfrak g$ be a finite-dimensional Lie algebra. Its symmetric Poisson algebra is denoted by $PS(\mathfrak g)$, and its universal enveloping algebra by $U(\mathfrak g)$. Canc…

Let $A$ be a simple locally finite associative algebra over $\mathbb{F}$, let $L=[A,A]$, and let $B$ be a Jordan-Lie inner ideal of $L$. Regularity conjecture. Every Jordan-Lie inn…

Makar-Limanov's conjecture. If $F\otimes_K A$ contains a free $K$-algebra on at least two free generators, then $A$ also contains a free $K$-algebra on the same number of free gene…

Countable-dimensionality conjecture. If $\tau$ is a locally finite Steinitz number, $A$ is a unital locally matrix algebra, and $\mathbf{n}(A)=\tau$, then $A$ is countable dimensio…

Non-finite generation conjecture. The operad ideal $\mathcal{K}$ is not finitely generated. Equivalently, no finite set of polynomial identities generates all identities satisfied…

Let $F$ be a field of characteristic different from two, and let $G(n)$ be the Grassmann algebra … A maximal commutative subalgebra is a commutative subalgebra maximal with respect…

Positselski's conjecture. Every such algebra $A$ satisfies

An associative shelf is a set equipped with an associative, self-distributive binary operation. Its two-term (rack) homology groups are the homology groups obtained from the associ…

Let $F$ be an algebraically closed field of characteristic zero. Let $A$ be a finite-dimensional basic algebra with Wedderburn–Malcev decomposition … where $J(A)$ is the Jacobson r…