255 problems

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…

Kashina's conjecture. The exponent of $H$ divides its dimension:

Let $H$ be a right noetherian Hopf algebra over a field, and let $S$ denote its antipode. Skryabin's bijective antipode conjecture. The antipode $S$ is bijective. Skryabin proved t…

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

Let $\operatorname{Qsym}[t_1,t_2,\dots]$ denote the algebra of quasi-symmetric functions in countably many variables over the integers. Ditters' conjecture. The algebra … is a free…

Let $H$ be a noetherian Hopf algebra over a field. Brown–Goodearl conjecture. Every noetherian Hopf algebra is Artin–Schelter Gorenstein. This conjecture concerns the homological r…

Regularity conjecture. If $H$ is finite-dimensional and cosemisimple, then $H$ is regular.

Wang–Zhang–Zhuang's classification conjecture. The families (I)–(VI) constitute all affine Hopf $k$-algebra domains of Gelfand–Kirillov dimension $2$, at least in the pointed case.

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

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.

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…

Let $F$ be a field, and let $\mathcal{H}^{\mathcal{M}}(F)$ and $\mathcal{H}^{\mathrm{f}}(F)$ denote the motivic and formal Hopf algebras of multiple polylogarithms. Quotient each b…

Classification conjecture. Every finite-dimensional simple partial $H$-comodule is of the form

A universal ring of invariants for algebraic structures of a fixed type is a polynomial algebra generated by certain trace diagrams. It admits the structure of a rational positive…

A pattern Hopf algebra is a Hopf algebra constructed from combinatorial objects equipped with a notion of restrictions; examples include permutation patterns, graphs, and marked pe…

Let $\Gamma$ be an abelian finitely generated group, let $V$ be a Yetter–Drinfeld module of diagonal type over $\Gamma$, and let…

Let $QSym(\mathbb Z)$ denote the universal ring of quasi-symmetric functions, identified with $J^{\vee}(\mathbb Z[y])=H^*(\Omega\Sigma\mathbb CP^{\infty};\mathbb Z)$, where $|y|=2$…

Let $\mathcal{C}$ be a finite tensor category over an algebraically closed field $\mathbf{k}$ of characteristic $q\ge 0$, with Frobenius–Perron dimension $p>2$ prime. A tensor cate…

Let $\mathfrak{g}$ be a complex semisimple Lie algebra, let $U_q(\mathfrak{g})$ be its quantised enveloping algebra, and let $W$ be the Weyl group of $\mathfrak{g}$. Consider the B…

Let $\widetilde{\operatorname{Cob}}^{2+1}$ be the category of 2-framed 3-dimensional relative cobordisms. Kerler's conjecture. The category $\widetilde{\operatorname{Cob}}^{2+1}$ a…

Let $H$ be a co-Frobenius Hopf algebra, meaning that it has a nonzero left integral (equivalently, any of the equivalent conditions characterizing co-Frobenius Hopf algebras). Its…

Let $(H,R)$ be a finite-dimensional triangular semisimple Hopf algebra, so that $R^{21}R=1$. Let $H_R\subseteq H$ denote its minimal sub-Hopf algebra. Gelaki's conjecture. The Hopf…

Let $M$ be the underlying vector space, and let $\tau=\triangle^\rho$ and $\rho=\triangle^\rho$ be the coproduct and product data determined by…

Khoroshkin–Tolstoy's conjecture. The double $DY(sl_n)$ is isomorphic to $\bar C$ as an algebra. The paper establishes this conjecture in the case of $DY(sl_3)$ by constructing a Ca…