1,076 problems

Let $U_{q}^{7}(\mathfrak{g})$ be an affine quantum group without the degree operator, and let $W^{r,s}$ be a Kirillov–Reshetikhin module, where $r$ is a node of the Dynkin diagram…

Let $A=(a_{ij})_{i,j\in I}$ be a symmetrizable Cartan matrix, let $\beta\in Q_n^+$, and let $\Lambda\in P^+$ be a dominant integral weight. Write $\mathscr{R}_{\beta}^{\Lambda}$ fo…

Let $e$ be the quantum characteristic of a non-trivial invertible element $q\in\mathbb{F}^{\times}$, and let $d_{\lambda,\mu}(v)$ be the integer-coefficient polynomials associated…

Let $\Gamma$ be the charge lattice of a four-dimensional $\mathcal N=2$ theory, with Dirac pairing $\langle\cdot,\cdot\rangle$, and let $\mathcal O(q)$ be the Kontsevich–Soibelman…

Let $S_N$ denote the classical symmetric group and let $S_N^+$ denote Wang's quantum symmetric group, with the canonical inclusion … The inclusion is conjectured maximality stateme…

Let $x\in\mathcal{X}$ and let $V\in \operatorname{Irr}\,\mathrm{U}^\imath_{v,x}$. Let $\varphi(x)\in G$ be the point associated with $x$, and let … be the $\theta$-twisted conjugac…

Berenstein–Zelevinsky conjecture. The product $b_1b_2$ belongs to the canonical basis up to a power of $q$ if and only if $b_1$ and $b_2$ $q$-commute.

Let … =^{-}oplusoplus^{+}. … , and let the diagram automorphism group of $^{+}$ have order $1$, $2$, or $3$. Andruskiewitsch–Dumas conjecture. The automorphism group of $U_q(^{+})$…

Berenstein–Zelevinsky's conjecture. Quantum double Bruhat cells admit a structure of quantum cluster algebras via quantum minors.

Let $L{A}$ be the subring of $\mathbb{C}(v)$ consisting of rational functions without a pole at $v=0$. Let $L{L}$ be the $L{A}$-submodule of $U^+_v(Lg{g})$ generated by the prescri…

Let ${\mathcal H}$ be the $\mathfrak{U}_q(\mathfrak{gl}_n)$-invariant Hamiltonian, and let ${\mathfrak b}_{(N-k)k}$ be a state in the $k$-sector satisfying … For…

Let the inclusive Racah matrices and the corresponding $\mathcal{R}$-matrices be those associated with three-strand knots, with the $\mathcal{R}$-matrix eigenvalues normalized by a…

Let $\mathfrak g$ be a simple Lie algebra of simply-laced type $\mathbb{A},\mathbb{D},\mathbb{E}$, and let ${\mathcal B}_q^*$ be the basis dual to the canonical basis of…

Quantum Gel'fand–Kirillov conjecture. The skew field $\operatorname{Fract} A$ is isomorphic to the skew field of fractions of a quantum affine space over $K$.

Let $M$ be a closed, oriented $3$-manifold, let $G$ be a reductive algebraic group, and let $\operatorname{Sk}_{G}(M)$ denote the vector space spanned by framed labelled graphs in…

Categorical braid group action conjecture. The functor $\cal{T}_{i,+1}'$ extends to an autoequivalence of $\textsf{Com}(\dot{\cal{U}}_Q(\mathfrak{g}))$ such that the induced automo…

Let $B^{r,s}$ be a Kirillov–Reshetikhin crystal associated with a Kirillov–Reshetikhin module $W^{r,s}$, where $r$ is a node of the affine Dynkin diagram other than $0$ and…

Let $W=(s_i\mid i\in I)$ be the Weyl group of $\mathfrak{g}$, let $\tau$ be the diagram involution, and let $\overline{I}_\tau$ denote the indices whose $\tau$-orbits are of finite…

Let $\mathcal{W}_p$ be the triplet VOA and let $\overline{U}_q(\mathfrak{sl}_2)$ be the restricted quantum group at a $2p$-th root of unity. FGST equivalence conjecture. The repres…

Let ${\mathfrak g}$ be a simple Lie algebra, and let ${{\rm KR}}_{{\alpha},i}(z)$ be Kirillov–Reshetikhin Yangian modules with pairwise distinct spectral parameters. For a tensor p…

Quantum-group interpolation conjecture. These standard combinatorial Hopf algebras should be incorporable into one-parameter families that contain an enveloping algebra and its dua…

Let $\mathfrak g$ be a Lie algebra with Cartan subalgebra $\mathfrak h$, and consider the De Concini–Procesi connection on the set of regular elements of $\mathfrak h$. Its horizon…

Let $L$ be an $ffrak{h}$-base manifold, let $P$ be an $ffrak{h}$-manifold, and let $pi$ be a Poisson dynamical bracket on $P$ over $L$. A quantization of the Poisson dynamical…

Let $\mathfrak h$ be the Lie algebra occurring in the classical dynamical Yang–Baxter equation, let $V$ be the representation space, and let a classical dynamical $r$-matrix be a m…

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…