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 $K=\Bbbk G$ 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 $F$ be a field of characteristic $0$, let $H$ be a Hopf algebra over $F$, and let $A$ be a finite-dimensional associative $H$-module algebra, with codimensions … where $P_n^H$…

Let $H$ be a semisimple Hopf algebra over $\mathbb{C}$. Kashina's conjecture. The exponent of $H$ divides $\operatorname{dim}(H)$. This conjecture is a divisibility analogue of the…

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.

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 Hopf algebra, let $B$ be a left coideal subalgebra, and let $(H/B^+H)^\ast\#B$ be the associated left partially dualized quasi-Hopf algebra. In Remark on invertibility…

Hopf algebra isomorphism conjecture. The map $\Upsilon$ induces a Hopf algebra isomorphism

Let $H$ be a connected Hopf algebra of finite Gelfand–Kirillov dimension, with antipode $S$. The Nakayama automorphism conjecture. The Nakayama automorphism of the coalgebra $H$ is…

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.

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.