21 problems

Let $(i,a)\in I\times_2\mathbb Z$ and $w\in W$, and let $Q_{w\varpi_i,a}$ be the corresponding formal solution of the extended $QQ$-system. Let $\chi_{\ell}$ denote the $\ell$-char…

Let $V_1,V_2,V_3$ be objects of the shifted category $\mathcal{O}_{sh}$. Associativity conjecture. There exists an isomorphism … Thus, $k$-fold tensor products in…

Let the affine Laumon space be the moduli space whose equivariant homology carries the relevant geometric representation, and let an iterated $W$-algebra of type $A$ and an affine…

Let $\boldsymbol{\boldsymbol{\text?}}$ be a tensor product parameter with associated free abelian group $S^{\boldsymbol{\boldsymbol{\text?}}}(V_{\mathbb Z})$, and let…

Extended QQ-system categorification conjecture. The module $V_w$ is of highest $\ell$-weight and fits into a non-split short exact sequence

Let $(i,a)\in I\times_2\mathbb Z$, let $w\in W$, and let $Q_{w\varpi_i^{\vee},a}$ be the corresponding $Q$-variable. Let $\chi_{\ell}$ denote the $\ell$-character and let…

Let $\mathcal{O}_{sh}^{\mathbb Z}$ be the integral finite-length subcategory of shifted category $\mathcal{O}_{sh}$, and let $\mathscr A$ be the cluster algebra associated with the…

Let $\mathfrak{r}$ be the group equipped with the involution $d$. The group-automorphism conjecture. The involution $d$ of $\mathfrak{r}$ is a group automorphism. This is proposed…

Let $\mu\in\mathbb{Z}_{\leqslant 0}$, and let $\varrho_\mu\colon Y_{\mu}(\widehat{\mathfrak{gl}}_1)\to \mathsf Y_{\mu}(Q,\mathsf{W})$ be the map from the shifted affine Yangian to…

Let $\tau$ be the diagram involution, let $\bar{\mathbf{v}}_j$ denote the associated class of the dimension vector at $j$, and let $\bm z$ be the central parameters. For a dominant…

Let ${\bf Z}=(Z_i(z))_{i\in I}$ be a truncation parameter. Define … where $u_{i,a}$ is the multiplicity of $a^{-1}$ as a root of $Z_i$. If $M$ is a monomial in $\chi({\bf Z})$, wri…

AII shifted twisted Yangian conjecture. A version of Theorems A and B should hold true for $beta=-1$ when $y_n^+(sigma)$ is replaced by the as-yet-undefined shifted twisted Yangian…

Let $\mathfrak{g}$ be the Lie algebra under consideration, let $\mu$ be the corresponding shift, and let $Y^{\mu}(\mathfrak{g})$ be the associative algebra defined by the RLL relat…

Let $\lambda$ be a dominant coweight and let $\mu\leq\lambda$ be a coweight such that $\mu+\alpha_i^\vee\leq\lambda$. Let $Y_\mu^\lambda(\mathbf{R})$ be the truncated shifted Yangi…

Let $Y_\mu^\lambda(\mathbf{R})$ be the image of the homomorphism $\Phi_\mu^\lambda(\mathbf{R})$. For each $i$, write $A_i(u)$ for the corresponding generating series, and let $m_i$…

Let $X$ be a smooth local toric Calabi–Yau $3$-fold, and let the Drinfeld double of its equivariant spherical Cohomological Hall algebra act on the cohomology of the moduli spaces…

Generalized-slice quantization conjecture. The truncations of $Y_{u}(gm])$ quantize the generalized slices in the affine Grassmannian.

Classical symplectic-leaf conjecture. The classical limit of the construction describes the full family of symplectic leaves in the Poisson-Lie group obtained as the classical limi…

Let $\mathfrak g$ be a simply-laced Kac–Moody algebra, and let $Y_\mu^\lambda$ be the corresponding truncated shifted Yangian equipped with its quotient filtration. Let…

Let $\rho$ be a cocharacter of $SL_N$, and use the combined $\mathbb C^\times$-action on $U^{comb}_\delta(\mathscr O_\epsilon(\mathbb C^2)\otimes\mathfrak{gl}_N)$ obtained from the…

Let $\overline{\Phi}_\mu^\lambda$ be the homomorphism defining the truncated shifted Yangian $Y^\lambda_\mu$, and let $A_i^{(p)}$ be the elements that vanish under…