408 problems

A graded Satake diagram is a decorated Dynkin diagram arising in the classification above, with the specified type I or type II $\mathbb{Z}_2$-graded Satake-diagram structure. Non-…

Renormalized r-matrix correspondence conjecture. Under the equivalence of Proposition E1, the renormalized $r$-matrix corresponds to the lowest non-trivial loop degree part of the…

Let $X_q$ be the half-quantum flag variety, let $\widetilde{\mathcal{N}}$ be the Springer resolution, let $\mathcal{N}$ be the nilpotent cone, and use…

Let $\widetilde{\mathcal{N}}$ be the Springer resolution, let $u(G_q)$ be the small quantum group, let $\operatorname{PrinBlock}_{\mathrm{dg}}(u(G_q))$ be its principal block, and…

Let $G$ be an almost-simple algebraic group with dual group $\check{G}$, let $q$ be a chosen complex root of unity, let $X_q$ be the half-quantum flag variety, let…

Let $\mathfrak{g}$ be a symmetrizable Kac–Moody algebra, let $W$ be its Weyl group, and let $w\in W$. Denote by ${\mathsf{A_q}}[N_{-}(w)]$ the corresponding quantum unipotent subgr…

Affine denominator conjecture. The affine Macdonald denominator is given by

Naturality and decomposability conjectures. The chain maps $\kappa_X$ and $\eta_X$ are natural, namely, they commute with all 2-morphisms of $\dot{\mathcal{U}}$. Moreover, for…

A Vassiliev invariant is a finite-type invariant of knots, and a quantum Lie group invariant is a knot invariant obtained from quantum-group representations. Bar-Natan's conjecture…

Let $\mathfrak{g}=\mathfrak{sl}(n)$, let $\tau$ be a Belavin–Drinfeld triple, and suppose that $r^0=t^0/2+\tilde{r}^0\in\mathfrak{h}\otimes\mathfrak{h}$ is $\tau$-admissible. Defin…

Let $(\Gamma_1,\Gamma_2,\tau)$ be an admissible triple and let $r^0\in\mathfrak h\otimes\mathfrak h$ satisfy the equations referenced in the source. Define $\tilde r^0=r^0-P^0/2$,…

Let $U_q(\widehat{\mathfrak{sl}_2})$ denote the quantum affine algebra. Let the twisted algebra be generated by $a_{1m}$ for $m\in 2\mathbb Z+1$, by $\gamma$, and by $x_{1n}^{\pm}$…

Faithfulness conjecture. The homomorphism $\kappa$ is faithful, that is, injective.

Let $U({\vec{z}},\tau,p)$ be the function defined in Corollary 1, namely … Here $u$ is the function appearing in the qKZB heat equation, $\alpha$ is multiplication by the function…

Let $\mathcal Q_n$ be the quantum permutation group on $n$ points, containing the classical symmetric group $S_n$. Maximality conjecture. The symmetric group $S_n$ is maximal in th…

Let $\vec\Lambda\in\mathbb Z_{\ge0}^n[2{\mathfrak m}]$, and let $\mathbb M_\Theta$ be the Macdonald–Ruijsenaars operator acting on $L_{\vec\Lambda}[0]$-valued trigonometric polynom…

Let $D$ be the determinant of the system of resonance relations, with parameters $x$ and $\Lambda_2,\dots,\Lambda_n$, and let $[z]$ denote the quantum-number factor used in the pap…

Let $\mathcal{C}$ be a hereditary finitary category. A derived functor is called friendly when it sends every object into two adjacent cohomological degrees. Automorphism conjectur…

Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be finitary hereditary connected $\mathbb{F}_q$-linear categories. An exact functor $F:D^b(\mathcal{C}_1)\to D^b(\mathcal{C}_2)$ is friendly…

Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be finitary hereditary connected $\mathbb{F}_q$-linear categories. Their reduced Drinfeld doubles are denoted by…

Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be finitary hereditary connected $\mathbb{F}_q$-linear categories, and let $\widetilde{\mathbf{D}}\mathbf{H}_{\mathcal{C}_i}$ denote their r…

Let $B_{\boldsymbol{\pi}'}$ and $B_{\boldsymbol{\pi}}$ be the crystals embedded as in equation, and let Theorem denote the stated agreement between the combinatorial realization an…

Let $(A,u)$ be a finitely generated quantum group, and let $b_k$ and $p_k$ denote the growth and random-walk quantities constructed in the paper. Here $f_k\approx g_k$ means that t…

Let $\mathcal{P}$ be the set of partitions, and define the partition functions … Here $f^\lambda$ is the dimension of the symmetric-group module indexed by $\lambda$. A singular po…

Let $\operatorname{Sing}_{n,q}^{\mathrm{per}}$ and $\operatorname{Sing}_{n,q}$ be the singular sets for the quantum permanent and quantum determinant, respectively. For…