42 problems

Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra, let $\mathrm{D}Y_\hbar(\mathfrak{g})$ be its Yangian double, and let $\boldsymbol{\mathcal{R}}$ be the un…

Let ${\mathcal M}$ be the space of quasi-triangular solutions to the classical Yang–Baxter equation, and let $r_{CG}(i,n)$ be the Cremmer–Gervais $r$-matrix indexed by positive cop…

Let $m$ and $n$ be relatively prime positive integers with $m<n$. Let $\mathfrak{p}_{m,n}\subseteq\mathfrak{sl}_n$ be the maximal parabolic subalgebra obtained by deleting the $m$-…

The Yangian Double has Cartan generators encoded by the operator-valued inverse $D^{-1}_{ij}(q)$, whose unitary expansion is defined symmetrically by … with the first term expanded…

Let $I_0$ index the fundamental representations, let $R^{{\operatorname{nor}}}_{ij}(x,y)$ be the normalized $R$-matrix for $i,j\in I_0$, and let $\gamma$ be the parameter defined i…

Let $I_0$ index the fundamental representations, let $i_1,\ldots,i_l\in I_0$, and let $a_1,\ldots,a_l\in\bar k^\times$, where $\bar k=\bigcup_{n>0}\mathbb C((q^{1/n}))$. Write…

Universal-twist evaluation conjecture. The TLA represented by a double arrow in the figures is conjectured to be the evaluation of a universal twist.

Universal-algebra lift conjecture. Whenever an analogous relation occurs at the level of evaluated $R$-matrices, a similar relation exists at the level of universal algebras.

Let U_{q,p}(chat{\bf sl}}_N) be the elliptic algebra in which the half currents and the $L$-operator $chat{L}^+(v)$ are defined, with spectral variables $z_i=q^{2u_i}$ for…

The q-Knizhnik–Zamolodchikov (qKZ) equations are q-difference equations whose connection matrices describe their q-monodromy. An elliptic quantum group is the appropriate elliptic…

Braided tangent-field conjecture. Equations (4) hold if we replace $U_i$ by $U_i^q$ and $f_{\beta,i}$ by $f_{\beta,i}(q)$.

Let $R$ be a unitary involutive $R$-matrix. An invertible matrix $\alpha$ is assumed to satisfy the relation referred to as Eq. (Ralphadef-graphical), and $\theta$ denotes the part…

In six-dimensional holomorphic Chern–Simons theory on twistor space, consider intersecting surface operators whose correlation function defines a local operator. A rational, spectr…

Let $\check{R}_{12}(\mu)$ be the R-matrix acting on sites $1$ and $2$, and let $C_1$ and $C_2$ be the corresponding local operators. Let $g_\pm(\mu)$ be functions of the spectral p…

Let $m,p\in 2\mathbb{Z}_{\geq 1}+1$, let $k<n$, and let $\mathfrak{g}=C_n^{(1)}$. Write $d_{k^m,k^p}(z)$ for the denominator polynomial of the corresponding normalized $R$-matrix,…

Sechin–Zotov commutation conjecture. The Hamiltonian $\mathcal{H}_2$ commutes with $\mathcal{H}_3$, that is, $[\mathcal{H}_2,\mathcal{H}_3]=0$.

Let $R$ be a quantum $R$-matrix, and let its eigenvalues be the spectral data associated with its action on the relevant tensor-product representation. Eigenvalue conjecture. Quant…

Let $V$ be a representation of $U_q\tilde{\mathfrak{g}}^\sigma$, and let $V(a)$ denote its spectral-parameter shift by $a$. Suppose that $V(a)\otimes V$ has a single non-trivial su…

Let $V(a)$ and $V$ be representations such that the tensor product $V(a)\otimes V$ has a single non-trivial submodule. The normalized $R$-matrix is denoted by $\check{R}^{V,V}(z)$.…

Let $\mathfrak{d}$ be the cotangent Lie algebra considered in the source, let $Y_1(\mathfrak{d})$ be its Yangian at parameter $1$, and let $\Delta_z$ be its meromorphic coproduct.…

Let $|\mu'\rangle=e^{(a)}_0|\mu\rangle$, where $|\mu\rangle$ is a state generated only from operators $e^{(a_i)}_0$. Let $\mathcal{T}_{\varnothing,\mu}(u)$ be given by … Here…

Let $\mathcal{T}_{\mu_1,\mu_2}$ denote matrix elements of the operator $\mathcal{T}$ between states labelled by partitions or related quiver data. In the known cases, for…

Let $X$ be the complete intersection of two quadric hypersurfaces in $\mathbb{CP}^5$, and let $\Sigma_2$ be a genus $2$ surface. Suppose an R-matrix relates the two splittings of t…

The Novikov peakon variables $(p_i,q_j)$ carry a quadratic Poisson structure, and one may seek a linear $r$-matrix structure directly associated to it, of the form … where $L$ is t…

Let $\mathsf{F}$ be the Fock space, let $\mathsf{RR}^{H}_{\mathsf{F},\mathsf{F}\otimes\mathsf{F}}$ denote the four-point operator in cohomological DT theory, and let…