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/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Let $Q$ be an acyclic quiver and $w \geq 1$ be an integer. Let$\mathsf{C}_{-w} (\mathbf{k} Q)$ be the $(-w)$-cluster category of $\mathbf{k}Q$. We show that there is a bijection between simple-minded collections in$\mathsf{D}^b (\mathbf{k} Q)$ lying in a fundamental domain of $\mathsf{C}_{-w}(\mathbf{k} Q)$ and $w$-simple-minded systems in $\mathsf{C}_{-w} (\mathbf{k}Q)$. This generalises the same result of Iyama-Jin in the case that $Q$ isDynkin. A key step in our proof is the observation that the heart $\mathsf{H}$of a bounded t-structure in a Hom-finite, Krull-Schmidt, $\mathbf{k}$-linearsaturated triangulated category $\mathsf{D}$ is functorially finite in$\mathsf{D}$ if and only if $\mathsf{H}$ has enough injectives and enoughprojectives. We then establish a bijection between $w$-simple-minded systems in$\mathsf{C}_{-w} (\mathbf{k} Q)$ and positive $w$-noncrossing partitions of thecorresponding Weyl group $W_Q$.

Functorially finite hearts, simple-minded systems in negative cluster categories, and noncrossing partitions