408 problems

Let $H$ be a quasi-triangular Hopf-$*$ algebra, and consider the corresponding Kitaev model on the plane. Haag duality conjecture. The model satisfies Haag duality for cones. This…

Tensor-product conjecture. The tensor product of regular positive representations is again regular, and it can be decomposed into a direct integral of regular positive representati…

Quantum cluster intersection conjecture. If $w_{1}\leq w_{2}$ in the Bruhat order, this triple intersection contains a quantum cluster algebra, and its quantum cluster monomials ar…

Let ${\cal M}$ be the set of multisegments. For ${\bf m},{\bf n}\in{\cal M}$, let $G^*({\bf m})$ and $G^*({\bf n})$ be the corresponding dual canonical basis elements, let…

Let $R_{\text{GGS}}$ be the matrix defined by … where the polynomials $P_{i,j,k,l}$ are uniquely determined by … Here $q=e^{\hbar}$, and the quantum Yang–Baxter equation is … while…

Slope subalgebra conjecture. The slope subalgebra $\mathcal{B}_{\mathbf{m}}$ is isomorphic to the quantum group

Let $m^{(i)}[p,s]$ be a Kirillov–Reshetikhin monomial, and let $F_q(m^{(i)}[p,s])$ and $L_q(m^{(i)}[p,s])$ denote the associated KR-polynomial and canonical-basis element. KR-polyn…

Let $\frak{S}$ be a triangulable punctured surface. A congruent ${\rm SL}_3$-lamination is an element $\ell \in \mathscr{A}_{{\rm SL}_3,\frak{S}}(\mathbb{Z}^t)$, and…

Braid group automorphism conjecture. For every such $i$, there exist mutually inverse algebra automorphisms $\mathbf T_{i,e}'$ and $\mathbf T_{i,-e}”$ of…

Kashiwara's conjecture. Every good finite-dimensional affine crystal, that is, every good crystal arising from a $U_q(\widehat{sl_n})$-module, is a tensor product of Kirillov–Reshe…

Let $L_\theta(\lambda)$ be the crystal lattice, let $c(L_\theta(\lambda))$ denote its bar-conjugate lattice, and let $V_\theta(\lambda)^{\mathrm{up}}_{\mathbf A}$ be the dual latti…

Let $V_\theta(\lambda)$ be the symmetric-crystal representation for a dominant integral weight $\lambda$ fixed by $\theta$, and let $L_\theta(\lambda)$ be its crystal lattice with…

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.