77 problems

Second Brauer–Thrall conjecture. Infinite representation type implies strongly unbounded representation type.

Let $A$ be an algebra, and let $\tau$-tilting theory be considered over a field $\mathbb{K}$. An algebra is of finite representation type when it has only finitely many isomorphism…

Let $\Lambda$ be a finite-dimensional $\mathbbm{k}$-algebra, and let $\mathcal{N}$ be the ideal of $HH^{\ast}(\Lambda)$ generated by all homogeneous nilpotent elements. Snashall–So…

Let $A$ be a finite-dimensional algebra over a field $K$, and let $\mathcal{N}$ be the ideal of $\operatorname{HH}^*(A)$ generated by all homogeneous nilpotent elements. Hochschild…

Let $A$ be a finite-dimensional algebra. Say that $A$ is brick-infinite if it admits an infinite family of pairwise non-isomorphic bricks, and brick-continuous if it admits an infi…

Let $n$ be a positive integer, and consider Hopf algebras of dimension $n$, with isomorphism as the equivalence relation. Kaplansky's tenth conjecture. In each dimension $n$ there…

Gorenstein bimodule conjecture. $A$ is self-injective if and only if $A$, regarded as an $A$-bimodule, is Gorenstein-projective.

First Brauer–Thrall conjecture. If $\Lambda$ is of infinite representation type, then it has indecomposable modules of arbitrarily large finite length.

Let $A$ be a finite-dimensional algebra and let $M$ be an $A$-module. The module $M$ is self-orthogonal if $\operatorname{Ext}^i_A(M,M)=0$ for every $i>0$. Enomoto's conjecture. Ev…

Vershik's conjecture. For every $n\geqslant 3$, there is a finite dimensional quadratic algebra with $n$ generators given by

Let $A$ and $B$ be finite-dimensional algebras over an algebraically closed field, and suppose they are stably equivalent, meaning their stable module categories are equivalent. Al…

Happel's conjecture. One has

Basic-Wild-Rank Conjecture. Let $A$ be a $d$-dimensional basic wild algebra. Then $r_A\leq10bd$.

Wild-Rank Conjecture. There is a function $f:\mathbb{N}\rightarrow\mathbb{N}$ such that $r_A\leq f(d)$ for all $A\in{\cal W}_d$.

Tame-Open Conjecture. For any $d\in\mathbb{N}_1$, all tame algebras in ${\cal A}_d$ form an open subset of ${\cal A}_d$.

Let $B_5$ be the braid group on five strands, let $Q(x)=x^3-\alpha x^2-\beta x-1$, and let … be the corresponding cubic Hecke algebra. The algebra $G_{32}$ is the finite group asso…

Let $\mathsf{A}$ be a finite-dimensional, indecomposable, unital commutative operator algebra of minimal quotient complexity. Here, minimal quotient complexity means that every $r$…

Converse conjecture. If every semibrick is left finite, then $A$ is brick finite.

Finitistic dimension conjecture. The finitistic dimension of $A$ is always finite.

First Finitistic Dimension Conjecture. One has

Let $A$ be a basic algebra, and suppose that $A$ is derived equivalent to a biserial fractional Brauer graph algebra (FBGA). Conjecture 1.3. Then $A$ is also a biserial FBGA. Since…

Let $\Lambda$ be a finite-dimensional algebra. A semibrick in $\operatorname{mod} \Lambda$ is a set of bricks whose distinct members have no nonzero morphisms between them. Enomoto…

Silting mutation conjecture. Then $T$ is tilting and $\operatorname{End}_A(T)$ is a $d$-representation infinite algebra.

Torsion-free Auslander–Reiten sequence conjecture. $A$ has $n$-torsion-free Auslander–Reiten sequences for every $n\in\mathbb{N}$ if and only if $A$ is self-injective.

First Brauer–Thrall conjecture. Infinite type implies unbounded type for the module category of $A$.