25 problems

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 $\Lambda$ be the coweight lattice, let $\Lambda^+\subset\Lambda$ be the dominant coweights, and let $\Psi_D$ be the homomorphisms defined in for $\Lambda$-valued divisors $D$ o…

Let $H_{+}$ be the subalgebra generated by $1,B_1,B_2,\dots$, and let $U_0$ denote the operator acting on the level-1 module under consideration. At the specialization $p=q^{2n}$,…

Let ${\cal L}(\Lambda_n)$ be the level $1$ integrable module of $A^{(2)}_{2n}$, and let $Y(B_n)$ be the Yangian of type $B_n$. Suppose … where $BS$ is the set of border strips with…

Let ${\cal L}(\Lambda_k)$ be the level $1$ integrable module of $\widehat{sl}_n$, and let the character decomposition in equation (26a) be the decomposition obtained from the irred…

Let ${^Cimath\mathscr{Y}}$ be regarded as a right coideal subalgebra of the Yangian ${\mathscr{Y}}$. For generating series … let $\xi_i(u)$ be the corresponding Yangian Cartan gene…

Let $Q=({\mathbb I},\Omega)$ be an ADE quiver, let $(V_i;W_i)_{i\in{\mathbb I}}$ be a representation of its framed double quiver, and impose the stated Satake-diagram symmetry and…

Let ${\mathcal A}$ be the algebra of local operators in a perturbative 3d holomorphic-topological QFT, and let ${\mathcal A}^!$ be its Koszul-dual $A_\infty$ algebra. A dg-shifted…

Let $\mathfrak{g}$ be a finite-dimensional differential graded Lie algebra and set … Let $\mathcal A_\hbar(\mathfrak{d},\gamma)$ be the Hopf algebra constructed in the source, and…

Let $Q$ be an arbitrary finite quiver and fix a dimension vector $\mathbf{d}\in\mathbf{N}^{Q_0}$. Let $\mathtt{a}_{Q,\mathbf{d}}(q)\in\mathbf{Z}[q]$ be Kac's polynomial, counting i…

Snake in the Snail conjecture. For odd $m$, the extended T-systems appear in successive lines of the snail operator, and the component corresponding to

Analytic-properties conjecture. The function $D_{1,\dots,m}$ is meromorphic in $\lambda_1,\dots,\lambda_m$, with at most simple poles when

Let $m,n,L$ be positive integers, let $\mathbf{M}=(M_1,\dotsc,M_{m+n-1})\in\mathbb{Z}_{\geq0}^{m+n-1}$, and let $\CM(\mathbf{M})$ be the Calabi–Yau target of the corresponding effe…

Let $J_{n+1}$ be the cactus group associated with the split real moduli space, and let the caterpillar points index the eigenbases of Bethe subalgebras on a tensor product of skew…

Let $X\in\overline{M_{0,n+2}}$ and let $B(X)$ be the corresponding Bethe subalgebra of $Y(\mathfrak{gl}_n)$. A representation of $Y(\mathfrak{gl}_n)$ is tame if it has the form…

Let $\mathcal{H}$ be a cyclic representation of the Yangian $\mathcal{Y}(\mathfrak{so}_{2r})$, with highest-weight vector a cyclic vector. Let…

Let $X$ be the local toric Calabi–Yau $3$-fold in the source, let $Y_\infty$ be the geometric Cartan doubled Yangian, and let $V:=H^*_c(\fM_\zeta^\chi,\varphi_W\mathbf{C})^\vee$ be…

The epimorphism conjecture. The map defined by

Let $Q$ be a quiver, let $\mathbf{g}_Q$ be the graded Lie algebra constructed from Maulik–Okounkov classical $R$-matrices, let $\bar{\mathfrak{g}}_Q$ denote the geometric Lie algeb…

Let $Q$ be a quiver, let $\mathbb{Y}_+$ be the positive half of the Maulik–Okounkov Yangian, let $K$ be the scalar extension used in the paper, and set…

Yangian degeneration conjecture. There is an isomorphism

Let ${\mathfrak{g}}$ be a semisimple Lie algebra, let $Y({\mathfrak{g}})$ be its Yangian, and let $M_{\mathfrak{g}}$ be the De Concini–Procesi compactification of the regular part…

Nekrasov–Shatashvili principle. The Baxter subalgebras of $\mathsf{Y}_Q$ corresponding to $g\in\mathfrak{H}$ are the algebras of modified equivariant quantum multiplication for Nak…

Let $\tilde{Y}^\lambda_\mu$ be the parameterized algebra obtained from $\tilde{Y}_\mu$ by base change along the map from the $r_i^{(s)}$ to the $c_i^{(s)}$ and imposing…

Integrable extension conjecture. The $\widehat{\mathfrak{sl}}_n$-module $V(\mu)$ extends to the irreducible integrable $\widehat{\mathfrak{gl}}_n$-module with highest weight $\mu$.