77 problems

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…

Let $Q$ be a quiver, let $\overline Q$ be its doubled quiver, and write … where $H^{\mathrm{BM}}$ denotes Borel–Moore homology. Let $Y_{MO}^{Q,+}$ denote the positive half of the M…

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…

Let $\bar{\rho}=\bar{\rho}_{\theta,\theta'}$ be the sign-twist-independent map denoted by $\Phi$, from the negative half $\mathbb{Y}_\mathcal{Q}^-$ of the Yangian to the equivarian…

Let $\mathfrak{gl}_{m|n}$ be the general linear Lie superalgebra, and let ${\rm Y}(\mathfrak{gl}_{m|n})$ be its Yangian. The formal power series $b_{m|n}^-(u)$, called the quantum…

Let the affine Laumon space be the moduli space whose equivariant homology carries the proposed representation, and let an iterated $W$-algebra of type $A$ and an affine shifted Ya…

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 $Q$ 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…

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…

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…

Khoroshkin–Tolstoy's conjecture. The double $DY(sl_n)$ is isomorphic to $\bar C$ as an algebra. The paper establishes this conjecture in the case of $DY(sl_3)$ by constructing a Ca…

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 $p=q^{2n}$,…

Kirillov–Reshetikhin reproduction-scheme conjecture. Every finite-dimensional representation of $Y({\mathfrak g})$ can be obtained from one specific representation by means of the…

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 $\kappa(\vec{h})$ be the border strip associated with the spectral datum $\vec{h}$, let $s^{\rm L}_{\kappa(\vec{h})}$ be its L-admissible tableau character, and let $Y(B_n)$ be…

Let $Z_N^{\rm P}(q,x)$ be the partition function of the $sl_n$ Polychronakos model, and define the vertex-model partition function by … Here ${\cal S}_N$ is the finite set of verte…

Let $V={\bf C}^n$, let $V^{\otimes N}$ be the spin space of the $sl_n$ Polychronakos model, and let $Y(sl_n)$ act on this space commuting with the Hamiltonian. Let the spaces $W_d$…

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 $U_q(\widehat{sl}_n)$ be the quantized affine algebra, and let the level $1$ integrable modules of the affine Lie algebra $\widehat{sl}_n$ carry their Yangian $Y(sl_n)$-module…

Finite-dimensional representation and classification conjecture. (i) $V$ can be lifted to an irreducible finite-dimensional $\mathcal D Y_{\hbar}(\mathfrak{sl}_N)$-module. (ii) The…

The paper considers the Lie algebra $\mathfrak{gl}_n(\mathbb{C})$, its universal enveloping algebra $U(\mathfrak{gl}_n(\mathbb{C}))$, and the polynomial families parameterized by d…

Drinfel'd-double Serre conjecture. The Serre relations in terms of modes have the same form as the six relations above, but now hold for every $k\in\mathbb Z$.

Mode-form Serre conjecture. For every $k\ge 0$, the supplementary Serre relations are

Universal-twist conjecture. These twist-like operators should be evaluation representations of universal twists obeying a shifted cocycle condition, so that the relation above is p…

Let $F_{N,n}$ be the Hilbert space of states of the Spin Calogero–Sutherland model with coupling parameter $\beta>0$, and let $H_{\beta,N}$ be its Hamiltonian. For integer non-nega…