English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the zaia usage as premium feature

ZAIA

Presents the pdf analysis as premium feature

PDF Analysis

Insights

Presents the summarisation as premium feature

Summarisation

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the access of premium content as premium feature

Premium Content

Zendy Tools

Zendy Open

We show that the category of finite-length generalized modules for thesinglet vertex algebra $\mathcal{M}(p)$, $p\in\mathbb{Z}_{>1}$, is equal to thecategory $\mathcal{O}_{\mathcal{M}(p)}$ of $C_1$-cofinite$\mathcal{M}(p)$-modules, and that this category admits the vertex algebraicbraided tensor category structure of Huang-Lepowsky-Zhang. Since$\mathcal{O}_{\mathcal{M}(p)}$ includes the uncountably many typical$\mathcal{M}(p)$-modules, which are simple $\mathcal{M}(p)$-module structureson Heisenberg Fock modules, our results substantially extend our previous workon tensor categories of atypical $\mathcal{M}(p)$-modules. We also introduce atensor subcategory $\mathcal{O}_{\mathcal{M}(p)}^T$, graded by an algebraictorus $T$, which has enough projectives and is conjecturally tensor equivalentto the category of finite-dimensional weight modules for the unrolledrestricted quantum group of $\mathfrak{sl}_2$ at a $2p$th root of unity. Wecompute all tensor products involving simple and projective$\mathcal{M}(p)$-modules, and we prove that both tensor categories$\mathcal{O}_{\mathcal{M}(p)}$ and $\mathcal{O}_{\mathcal{M}(p)}^T$ are rigidand thus also ribbon. As an application, we use vertex operator algebraextension theory to show that the representation categories of all finitecyclic orbifolds of the triplet vertex algebras $\mathcal{W}(p)$ arenon-semisimple modular tensor categories, and we confirm a conjecture ofAdamovi\'{c}-Lin-Milas on the classification of simple modules for these finitecyclic orbifolds.

Ribbon tensor structure on the full representation categories of the singlet vertex algebras