9 problems

Let $W^{(j)}(z)$ be the currents of the deformed $W_n$ algebra (DWA), and let ${\overline X}_{\alpha}(\lambda)$ denote the screening operators defined by (SET1). An intertwiner is…

A vertex algebra $\mathcal{V}$ has nonlocal screening operators arising from modules $\mathcal{M}\neq \mathcal{V}$; these operators may involve non-integral powers of the integrati…

Narrow-screening conjecture. For every $\lambda\in\Lambda$ and $\sigma\in W$, $F_{\sigma,\lambda}$ is independent of the choice of a minimal expression of $\sigma$. Moreover,

Let a rank-two Cartan $q$-diagram and corresponding $m$-solution be given, with parameter $r>0$ and screening operators $\mathfrak{Z}_1$ and $\mathfrak{Z}_2$. The simple truncation…

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 $zem_{\alpha_i^{long}}$ be the long screening operators associated with the long basis of the integral lattice, and let $\mathfrak{g}$ denote the corresponding Lie algebra (or…

Let ${\cal F}_{mn}$ and ${\cal F}^{\cal A}_{mn}$ be level-graded spaces, and let the right action of $\Sigma$ define complexes because $\Sigma^2=0$. Consider the cohomology at the…

Maximal-rank conjecture for $W$. For generic values of $r$, these right actions are surjective if $m-n\le -2$ and indegenerate if $m-n\ge -2$; equivalently, the operator $W$ has ma…

Let $L=\sqrt{p}Q$ with $p\geq 2$, let $\mathfrak{g}$ be simply laced of rank $l$, and let $\mathcal{W}_k(\mathfrak{g})$ be the associated principal W-algebra with $p=k+h^\vee$. For…