22 problems

Let $A_{w_1,\ldots,w_m}$ be a Lance–Tapper -algebra, meaning that it is generated by $x_1,\ldots,x_n,x_1^*,\ldots,x_n^*$ with reduced relations $w_j=0$ and $w_j^*=0$, and every def…

Two algebras are called similar in the sense of Schack when they satisfy the notion of similarity defined in the cited work. Schack's similarity conjecture. Algebras that are simil…

Let $\mathfrak{O}$ be the coefficient ring and let $A$ be a finite-dimensional $\mathfrak{O}$-algebra. Write $\zeta_A(t)$ and $\zeta^{\lhd}_A(t)$ for its subalgebra and ideal zeta…

Let $\Lambda$ be a finite representation type algebra, and let $\widetilde{\mathcal{U}}_\Lambda$ and $\widetilde{\mathcal{M}}_\Lambda$ be the affine varieties associated with…

Let $A$ be a connected $\mathbb{R}$-algebra. An algebra is locally complexified-gentle of uniform type if it is Morita equivalent to an algebra satisfying condition (1) or (2) of T…

Let $A$ be an indecomposable algebra in the tensor category under consideration. Etingof's conjecture. The algebra $A$ is exact if and only if it is simple. This conjecture relates…

Rickard's conjecture. Any derived equivalence between flat algebras is standard.

Canonical derived equivalence conjecture. Any derived equivalence

Connectedness conjecture. If there is a path between $A$ and $0$ in $\mathcal{H}(\operatorname{\mathsf{s\tau-tilt}} A)$, then $\mathcal{H}(\operatorname{\mathsf{s\tau-tilt}} A)$ ha…

In a solvable Interaction Round the Face lattice model, let the Boltzmann weights satisfy a polynomial equation of degree $n$. The associated algebra is called the $n$-CB algebra.…

Two-simple-module conjecture. Up to a trivial glueing procedure, every geometrically irreducible algebra has, up to isomorphism, at most two simple modules.

Let $A$ be a piecewise hereditary algebra of tree type, let $\hat{A}$ be its repetitive category, and let $\phi$ be an automorphism of $\hat{A}$ with nonzero jump $n$. Write … for…

Let $S$ be a finitely presented monoid of linear growth, and let ${\mathbb F}_2 S$ be its monoid algebra over the field with two elements. An algebra is automaton if its language o…

Derived dimension conjecture. The derived dimension of $A$ satisfies

Bridge-free matroid criterion. The algebras $\mathcal C_{G_1}^T$ and $\mathcal C_{G_2}^T$ are isomorphic if and only if their bridge-free matroids are isomorphic.

Basis conjecture. The spanning set of Proposition $$ constitutes a $\mathbbm{k}$-basis for $\mathcal{H}^{\mathbf{C}}_n(A)$.

Let $\cA$ be a non-associative $\fo$-algebra that is free of rank $d$ as an $\fo$-module, where $\fo$ is the ring of integers in a number field $k$. Root-location conjecture. If…

Let $\cA$ be a non-associative $\fo$-algebra that is free of rank $d$ as an $\fo$-module, where $\fo$ is the ring of integers in a number field $k$. Zero-pole conjecture. The topol…

Let $\cA$ be a non-associative $\fo$-algebra that is free of rank $d$ as an $\fo$-module, where $\fo$ is the ring of integers in a number field $k$. Degree conjecture. … This is pr…

Let $\Lambda$ be a finite dimensional $k$-algebra, let $(U,P)$ be an almost complete support $\tau$-tilting pair, and let $(X,0)$ and $(Y,0)$ be two complements. After interchangin…

Let $A$ and $B$ be algebras, and let $X$ and $Y$ be $A$-$B$-bimodules. A bimodule homomorphism $f:X\to Y$ determines the 2-diagram described by the maps involving $Z(A)$, $Z(B)$, a…

Let $\mathbb{R}^8$ be equipped with the commutative multiplication defined by $\phi_i$. Simplicity conjecture. The algebra $\mathbb{R}^8$ with this commutative multiplication is si…