23 problems

Let ${\mathfrak{g}}$ be a simple Lie algebra, let $\lambda=\lambda_1+\cdots+\lambda_n$ be a sum of dominant integral weights, and let $c_1,\ldots,c_n$ be pairwise distinct complex…

Let $\mathfrak{g}$ be a simple Lie algebra, let $k\in\mathbb{Z}_{>0}$, and let $\lambda_s\in P^+$ for $1\leq s\leq k$. For pairwise distinct complex numbers $a_1,\ldots,a_k$ and pa…

Let ${\mathfrak g}$ be a simple Lie algebra with fundamental representations $V^{(i)}$, $i=1,\dots,r$, and highest weight vectors $v^{(i)}\in V^{(i)}$. Let…

Let $P^+$ be the set of dominant integral weights, and let $V(\gamma)$ denote the irreducible highest weight representation of highest weight $\gamma$. For pairs of dominant weight…

Bracket-width conjecture. The bracket width of $\mathfrak g\otimes_{G_a}G_a[[t]]$ is equal to $2$ if $\mathfrak g$ is of type $\mathrm A_n$ or $\mathrm C_n$ $(n\ge 2)$, and to $1$…

Let $W_1,\dots,W_r$ be the modules considered in the preceding construction, let $\mathcal{A}(\lambda_1,\dots,\lambda_r)$ be the associated parameter algebra, and let…

Let $P^+$ be the dominant integral weights of $cmathfrak{sl}_n$, let $cmathcal{P}_{clambda}$ be the set of decompositions of $clambdain P^+$ into two dominant weights, and let…

Let $cmathfrak{g}$ be a semisimple complex Lie algebra, let $P^+$ denote its dominant integral weights, and fix $clambda_1,clambda_2in P^+$. Put $clambda=clambda_1+clambda_2$, and…

Let $cmathfrak{g}$ be a semisimple complex Lie algebra, let $P^+$ be its dominant integral weights, and let $a_1,\ldots,a_sincmathbb C$ be pairwise distinct with…

Let $\mathcal A_q$ be the current algebra, let $\mathcal O_q$ be the q-Onsager algebra with generators $A,B$, let $\{\tilde g_{k+1}\}_{k\in\mathbb N}\subseteq\mathcal J$ be the seq…

Let $\mathcal O_q$ be the q-Onsager algebra with generators $A,B$, and let $\mathcal J$ be its two-sided ideal generated by $A$ and $B$. Lower-generator uniqueness conjecture. Ther…

Let $J$ be the two-sided ideal of $\mathcal A_q$ generated by $\mathcal W_0$ and $\mathcal W_1$, and let $Z(\mathcal A_q)$ denote the center. Ideal–center direct-sum conjecture. Th…

Let $\mathcal A_q$ be the current algebra, let $Z(\mathcal A_q)$ denote its center, let $\Delta_{k+1}\in Z(\mathcal A_q)$, and let $\langle\mathcal W_0,\mathcal W_1\rangle$ be the…

Let $\mathcal A_q$ be the current algebra over $\mathbb F$, with generators $\mathcal W_{-k}$, $\mathcal W_{k+1}$, $\mathcal G_{k+1}$, and $\mathcal{\tilde G}_{k+1}$ for…

Integral trace conjecture. We have

Let $\mathbf{g}$ be a finite-dimensional Lie algebra, and let $\mathcal{I}(\mathbf{g}\otimes\mathbb{C}[x])$ denote the category of graded finitely generated $\mathbf{g}$-locally fi…

Let $\lambda\in P_0^+$, where $P_0^+$ denotes the set of dominant weights relevant to the twisted current algebra, and let a graded local Weyl module and the associated graded modu…

Let $\lie g$ be a simple Lie algebra, let $P^+$ denote its dominant integral weights, and let $M(\lambda)$ and $T(\lambda)$ be the graded $\lie g[t]$-modules defined in the paper f…

Suppose that $\mathfrak{g}$ is of type $B_n$, $C_n$ or $D_{n+1}$ and that $\lambda\in P^+$ satisfies $\lambda(h_i)=0$ for $i\ge n$. Let $\Psi_\lambda$ be the root subset defined fr…

Let $A=\mathcal R_{k,\ell}$ for some $k,\ell\in\mathbb Z_+$. For $\bos=(s_1,\ldots,s_n)\in\mathbb Z_+^n$, let $W_A(\bos)$ denote the corresponding global Weyl module, and let $P^+$…

Let $\lie g$ be of type $D$ or $E$, with the decomposition used in the paper into subdiagrams $I_1$, $J$, and $I_2$, where $J$ is of type $D_4$. Let $\lambda\in P^+$ and let…

Let $\lie g$ be of type $D$ or $E$, let $i_0$ be its unique triply connected node, and let $\lambda\in P^+$ be supported on the three connected components of $I\setminus\{i_0\}$. F…

Let $\lambda\in P^+$ be such that its extended support $\overline{\operatorname{supp}}(\lambda)$ does not contain a subdiagram of type $D_4$. Let…