32 problems

Let $Mag(V)$ be the free magmatic algebra on a vector space $V$, and let $NAP^c(V)$ denote the corresponding cofree $NAP^c$-coalgebra. The coalgebra map … is the canonical quotient…

A Lie-coalgebra–PostLie bialgebra is the conjectural bialgebra structure obtained by combining a Lie coalgebra with a PostLie algebra; a free PostLie algebra is the free algebra fo…

Noncommutativity conjecture. The homology bialgebra of the dual of ${\mathcal {T}_*\mathcal{D}}^{odd}$ is not commutative for some ring of coefficients.

Let $\mathcal{H}$ be the bialgebra object, let $\mathcal{H}_{\infty}$ be the canonical differential graded submodule resolving its prematrad version…

Tensor-product non-isomorphism conjecture. In general, the bialgebras

Deformation-theoretic conjecture. There exists a deformation theory for $A_{\infty}$-bialgebras in which the infinitesimal deformations of classical differential graded bialgebras…

Dual connecting-functor conjecture. If

Let $m,n$ be positive integers, let $\xi^m_n$ denote the generator indexed by $m$ inputs and $n$ outputs, and let $\partial$ be the perturbed differential in the minimal model of t…

Let $A$ be the complex of bracket star-diagrams, with its differential bialgebra structure, and let $H_*(\mathcal K)$ denote the homology bialgebra of the space $\mathcal K$ of lon…

Let $d$ be the ambient dimension, let $A$ be the complex of bracket star-diagrams equipped with the differential bialgebra structure constructed in the paper, and let…

Let $H$ be a bialgebra and let $\mathbf R=\{R^{(c)}\}_{c\in C}\in H\widehat\otimes H$ solve the transitive quantum Yang–Baxter equation … whenever $c'\in\{c,c”\}\subset C$. For an…

Transitivity conjecture. The element $\mathbf R^{(\boldsymbol{\epsilon})}$ solves the QYBE. This is expected to provide a large noncommutative class of QYBE solutions analogous to…

Conjecture. Integrals in $B^*$ can be extended to integrals in $\mathrm{H}(B)^*$.

Let $G,G',H,H'$ be compact Lie monoids, let $X,X'$ be closed smooth manifolds acted on by $G\times H$ and $G'\times H'$ respectively, let $\varphi\colon G\to G'$ and…

Fix $\alpha\geq1$, and let $\mathcal{A}_{\Pi_{(A_{2},\alpha)}}$ be the graded algebra associated with the preprojective construction for the quiver of type $A_{2}$ with multiplicit…

Let $A$ be a unital Jordan algebra, and consider trigonometric and quasi-trigonometric solutions of the $A$-classical Yang–Baxter equation (A-CYBE). Jordan unitality obstruction co…

Let $A$ be a unital non-associative strongly geometrically admissible algebra, and consider solutions of the $A$-classical Yang–Baxter equation (A-CYBE) of trigonometric or quasi-t…

Let $(A,\Phi)$ and $(A',\Phi')$ be dual $\ell$EAS, and let $B_{\Phi,(\mathcal{D},\times)}$ and $B_{\Phi',(\mathcal{D},\times)}$ denote the associated bialgebras built from the same…

Let $A$ be a primitive bialgebra in a braided monoidal abelian category, and let ${\mathbb{B}\mathbb{B}}_n(A)$ denote its $n$th bibar-cube, viewed as a double representation of…

Let $\sim$ be exotic Knuth equivalence, let $\mathbb{K}_\textsf{P}^{(\sim)}$ be the associated bialgebra, and let $\tilde\psi_{>|\leq}$ be the associated map. Exotic Knuth Schur-po…

Let $\sim$ be exotic Knuth equivalence, let $\mathbb{K}_\textsf{P}^{(\sim)}$ be the associated bialgebra, and let $\tilde\psi_{>|\leq}$ be the map to the completion of peak quasi-s…

Quiver zastava bialgebra conjecture. The bialgebra $\mathcal A^Q$ is isomorphic to

Schieder's conjecture. The bialgebra $\mathcal A$ is isomorphic to the universal enveloping algebra

Let $B$ be a conilpotent dg bialgebra, and let $C_{GS}^{\mathrm{full}}(B,B)$ be its full Gerstenhaber–Schack complex. Full Gerstenhaber–Schack deformation conjecture. The underlyin…

Let $V$ be a vector space, and let $\operatorname{Sym}(V)$ be its symmetric bialgebra. Write $\operatorname{Def}(\operatorname{Sym}(V))$ for the deformation complex of this symmetr…