45 problems

Refined skein recursion conjecture. The equation obtained for the refined Chern–Simons model is the correct refinement of the skein recursion relations for the unknot.

Universal formula conjecture. The formula

Let $L(\overline{m_{1,2n}})$ be the indicated simple $\mathcal{U}_q({\mathfrak{sl}_{2n,\mathrm{tor}}})$-module, and let $\ast$ denote the fusion product. Imaginary-module conjectur…

Let $\phi_n$ be the homomorphism folding $q$-characters of integrable $\mathcal{U}_q({\widehat{\mathfrak{sl}_{\infty}}})$-modules to virtual $q$-characters of integrable…

Let three D3 branes be represented by three vector representations of $U_{\mathfrak{q},\mathfrak{t}}(\widehat{\widehat{\mathfrak{gl}}}_1)$, and consider the coproduct composition…

Let $M,N_1,N_2\in\mathbb{Z}$. The algebras $\mathcal{A}_{M,N}$ contain a subalgebra $\mathcal{K}_{M,N}$ extending the commuting quantum toroidal $\mathfrak{gl}_1$ algebras, and…

Refined torus skein conjecture. The refinement of the skein algebra of the torus is the quantum toroidal $cmathfrak{gl}_1$ algebra.

Quantum toroidal superalgebra isomorphism. There is an isomorphism

In the level $N$ representation of the quantum toroidal $\mathfrak{gl}_1$ algebra, the relevant $q$-deformed conformal blocks of two-dimensional conformal field theories are compar…

Eigenvector–oper correspondence conjecture. For every $l_0\in\mathbb Z_{\ge0}$, there is a one-to-one correspondence between joint eigenvectors of $\mathcal A({p}_{0},{p}_{1})$ on…

Elliptic cohomological realization conjecture. The representation in Theorem gives the level-$(0,1)$ irreducible representation of $U_{q,\kappa,p}(\mathfrak{gl}_{N,\mathrm{tor}})$…

D6 $qq$-character BPS-crystal conjecture. For $i\in Q_0$, the $qq$-characters are

Gauge-origami Bethe ansatz conjecture. The Bethe ansatz equations are

The quantum toroidal $\mathfrak{gl}_1$ algebra admits a MacMahon representation, in addition to the Fock, vector, and other representations used in intertwiner constructions. Plane…

Affine ADE geometric quantum group conjecture. The associated geometric quantum group should be isomorphic to the corresponding quantum toroidal algebra of ADE type.

Hopf algebra isomorphism conjecture. The map $\Upsilon$ induces a Hopf algebra isomorphism

R-matrix correspondence conjecture. The endomorphism

Let the matrix analog of the Miura transformation be a matrix generalization describing the rectangle $W$ algebra, and let its q-deformation be the corresponding q-deformed constru…

Let $q_1,q_2,q_3$ be deformation parameters, let $T_i(z)$ be the higher currents, and let $f_{i,j}(z)$ be the structure functions defined for $i\leq j$ by … and set…

Let $\ddot{\mathrm{U}}_q^0(\mathfrak a_1)$ denote the degree-zero subalgebra of the double quantum affinization algebra of type $\mathfrak a_1$, and let…

Let $\mathcal{K}$ be the algebra acting on the tensor product module … where $u_1,\dots,u_\ell$ are parameters. Irreducibility conjecture. The $\mathcal{K}$-module $\mathbb{F}$ is…

Let $\mathcal{F}_{\Lambda}$ be the Fock representation with screening operators $\xi_i$ and $\eta_i$ for $i\in\hat{I}^-\cup\{m\}$, and set … For a weight $\Lambda$, let…

Let $U_q\,\widehat{\mathfrak{sl}}_{m|n}$ be the quantum affine algebra and let ${\mathcal E}_{m|n}$ be the quantum toroidal algebra. Define…

Let $\mathcal E_{q^{-4},q^2,q^2}$ be the specialized elliptic Hall algebra, let $\widehat{\mathcal E_{q^{-4},q^2,q^2}}$ denote its completion, and let…

Let $\mathcal E_{q_1,q_2,q_3}$ be the elliptic Hall algebra with $q_1q_2q_3=1$, and let $\mathcal E_{q^{-4},q^2,q^2}$ be its specialization at $(q^{-4},q^2,q^2)$. Let…