22 problems

Schneider's conjecture. The constructed solution coincides with the GGS solution.

Let a homology group carry the structure of an involutive Lie bialgebra, a particular type of Lie bialgebra. Lifting conjecture. This structure can be lifted to a structure of an i…

Necessity conjecture. All solutions of the classical Yang–Baxter equation are described by the preceding ansatz and differential-equation condition. The statement is presented as t…

Let $W_n$ be the Lie algebra of formal vector fields on ${\mathbb R}^n$, and consider coboundary triangular Lie bialgebra structures on $W_n$, equivalently coboundary Poisson–Lie s…

Let $\mathfrak{g}$ be a Lie bialgebra with cobracket $\delta$, let $U(\mathfrak{g})$ denote its universal enveloping algebra, and let $A^{!}_{\mathcal{N}}$ be the $E_1$ Koszul dual…

Let $n$ be a positive integer, let ${\mathfrak g}$ be a Lie algebra, and let $r\in {\mathfrak g}\otimes {\mathfrak g}$ solve the classical Yang–Baxter equation (CYBE). Let…

Let $Q$ be a manifold and let $W$ be a Liouville domain. Write $\widehat{\mathbb{H}}_*^{S^1}\Lambda$ for Rabinowitz string homology, $S\mathbb{H}_*^{S^1}({\partial}W)$ for $S^1$-eq…

Finite-dimensional quantization conjecture. The statement that every universal quantization is homotopy equivalent to a direction-involution-invariant universal quantization is not…

The properads $\mathcal{H}\mathit{olieb}$ and $\mathcal{L}\mathit{ieb}$ control homotopy Lie bialgebra structures, while their wheeled closures…

Let $\mathrm{GC}_2$ be the graph complex governing the obstructions to a universal iterative deformation-quantization procedure for Lie bialgebras. Its degree-one cohomology is den…

Let $\widehat{W}_N$ be the graded vector space generated by cyclic words in the letters $\{x_1[-p],\ldots,x_N[-p]\}_{p\in\mathbb{N}}$, and let … be the explicitly constructed stron…

Torsor conjecture. The set of homotopy classes of such morphisms is a torsor over the Grothendieck–Teichmüller group $GRT$. The conjecture predicts that the explicit transcendental…

A Lie bialgebra is a Lie algebra equipped with a compatible Lie cobracket, and its deformation quantization is expected to be governed by a homotopical structure analogous to the o…

KKPS conjecture. The cohomology set $H_{BD}^1({\bf G},r_{\rm DJ})$ is trivial.

A Lie bialgebra is a vector space with a Lie bracket and a compatible Lie cobracket. Drinfeld's quantization conjecture. Any Lie bialgebra can be quantized. This conjecture asks wh…

Let $r_{BD}$ be an $r$-matrix from the list in Lemma, and let $\mathfrak{g}$ be the corresponding Lie algebra. Twisted Lie bialgebra structures on $\mathfrak{g}$ corresponding to…

Koszulness conjecture. The properad ${\mathcal L}ie{\mathcal B}^{\diamondsuit}$ is Koszul.

Interpretation conjecture. Symmetric quasi-invariant tensors in differential crossed modules, in addition to describing infinitesimal 2-braidings, can be interpreted in the setting…

Let $\mathfrak{g}$ be a simple Lie algebra, let $G=\operatorname{Ad}(\mathfrak{g})$ be its adjoint group, and let $r_{DJ}$ be the Drinfeld–Jimbo $r$-matrix. For a Belavin–Drinfeld…

Let $\mathfrak{g}$ be the Lie algebra occurring above, let $\mathfrak{g}[u]$ be its polynomial current algebra, and for non-zero constants $c_1\neq c_2$ let … These $r$-matrices de…

Homology conjecture. The homology of the properad $\operatorname{\Omega}(\mathrm{co}\mathbf{Frob})$ is the involutive biLie properad.

Cobar resolution conjecture. The properad $\operatorname{\Omega}(\mathrm{co}\mathbf{Frob})$ gives a resolution of the Lie bialgebra properad.