93 problems

For every finite-dimensional semisimple and cosemisimple Hopf algebra $H$ over a field $k$, the exponent of $H$ divides its dimension: $\exp(H)\mid\dim_k H$.

Let $\mathcal{B}_{\mathbf{0}}^{+}$ be the slope subalgebra of the preprojective $K$-theoretic Hall algebra, and let $\mathfrak{n}_{Q}^{BPS}$ denote the positive half of the BPS Lie…

Let $K$ be an algebraically closed field of characteristic zero, let $G$ be a finite abelian group, and let … Let $A$ be a finite-dimensional cocommutative cosemisimple Yetter–Drin…

Auslander's conjecture. The skew group ring $A\#G$ is naturally isomorphic to

Faithfulness conjecture. The homomorphism $\kappa$ is faithful, that is, injective.

Let ${\cal H}$ be the Heegaard category, and let $T(V)$ denote the relevant target associated with a vector space $V$. Residual representability conjecture. If $a$ and $b$ are dist…

Let ${\bf PQSym}^*$ denote the dual Hopf algebra of parking functions, realized as a dendriform trialgebra with operations induced from the dendriform trialgebra structure on…

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…

Cyclic cohomology conjecture. The cyclic cohomology of $(H,R)$ satisfies

Center-trace equality conjecture.

Universality conjecture. The formal group ${\frak F}(x\otimes 1,1\otimes x)$ is the universal object in the category of formal groups over a (topological) Hopf algebra.

Unipotency conjecture. The operator

Let $\mathcal{T}$ be the residual symmetry algebra, let $\mathcal{U}$ be the non-confined algebra, and let $\Gamma:\mathcal{T}\to\mathcal{U}$ be the Hopf map. Define the left and r…

Let $G$ be a finite group. Say that $G$ collapses if every finite-dimensional pointed Hopf algebra $H$ with group of grouplikes $G(H)\simeq G$ is isomorphic to $\Bbbk G$; equivalen…

Expected implication. If $H$ has a $\Phi$-set theoretic basis, then $H$ has the positive basis property.

Let $M,N_1,N_2\in\mathbb{Z}$. The algebras $\mathcal{A}_{M,N}$ contain a subalgebra $\mathcal{K}_{M,N}$ extending the commuting quantum toroidal $\mathfrak{gl}_1$ algebras, and…

Classification conjecture. Up to equivalence, the only bicompatible permutation statistics are the descent set $\operatorname{Des}$, the peak set $\operatorname{Pk}$, the valley se…

Lifting conjecture. The set $X$ is a complete set of exact $H$-equivariant Morita equivalence class representatives.

Let $K=\Bbbk G$ for a finite group $G$, and let … be a finite-dimensional coradically graded connected Hopf algebra, with $R^1$ its degree-one component. Andruskiewitsch–Schneider…

Let $\widetilde{\mathcal{P}}(S^{n-1})$ be the reduced spherical scissors congruence group, and let the direct sum … be equipped with the Sah algebra Hopf structure, whose coproduct…

Dixmier–Moeglin conjecture for pointed Hopf algebras. The following conditions are equivalent:

Let $H$ be a finite-dimensional Hopf algebra acting on a Weyl Poisson algebra. Rigidity conjecture. The $H$-action must factor through a group algebra. This conjecture asks whether…

Categorical splitting conjecture. There is an equivalence of braided tensor categories

A noetherian Hopf algebra is a Hopf algebra satisfying the ascending chain condition on left and right ideals. Brown-Goodearl conjecture. Every noetherian Hopf algebra is Artin–Sch…

Let $H$ be a finite-dimensional quasitriangular Hopf algebra and let $\tilde H$ be its formal ribbon extension. Let $M_1,M_2$ be indecomposable $H$-modules, and let…