Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2411.11633 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909931801673728 |
|---|---|
| author | Grabowski, Jan E. Pressland, Matthew |
| author_facet | Grabowski, Jan E. Pressland, Matthew |
| contents | We develop a general theory of cluster categories, applying to a 2-Calabi-Yau extriangulated category $\mathcal{C}$ and cluster-tilting subcategory $\mathcal{T}$ satisfying only mild finiteness conditions. We show that the structure theory of $\mathcal{C}$ and the representation theory of $\mathcal{T}$ give rise to the rich combinatorial structures of seed data and cluster ensembles, via Grothendieck groups and homological algebra. We demonstrate that there is a natural dictionary relating cluster-tilting subcategories and their tilting theory to A-side tropical cluster combinatorics and, dually, relating modules over $\underline{\mathcal{T}}$ to the X-side; here $\underline{\mathcal{T}}$ is the image of $\mathcal{T}$ in the triangulated stable category of $\mathcal{C}$. Moreover, the exchange matrix associated to $\mathcal{T}$ arises from a natural map $p_{\mathcal{T}}\colon\mathrm{K}_0(\operatorname{mod}\underline{\mathcal{T}})\to\mathrm{K}_0(\mathcal{T})$ closely related to taking projective resolutions.
Via our approach, we categorify many key identities involving mutation, g-vectors and c-vectors, including in infinite rank cases and in the presence of loops and 2-cycles. We are also able to define A- and X-cluster characters, which yield A- and X-cluster variables when there are no loops or 2-cycles, and which enable representation-theoretic proofs of cluster-theoretical statements.
Continuing with the same categorical philosophy, we give a definition of a quantum cluster category, as a cluster category together with the choice of a map closely related to the adjoint of $p_{\mathcal{T}}$. Our framework enables us to show that any Hom-finite exact cluster category admits a canonical quantum structure, generalising results of Geiß--Leclerc--Schröer. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_11633 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cluster structures via representation theory: cluster ensembles, tropical duality, cluster characters and quantisation Grabowski, Jan E. Pressland, Matthew Representation Theory Quantum Algebra Rings and Algebras 13F60 (Primary), 14T10, 18F30, 18N25 (Secondary) We develop a general theory of cluster categories, applying to a 2-Calabi-Yau extriangulated category $\mathcal{C}$ and cluster-tilting subcategory $\mathcal{T}$ satisfying only mild finiteness conditions. We show that the structure theory of $\mathcal{C}$ and the representation theory of $\mathcal{T}$ give rise to the rich combinatorial structures of seed data and cluster ensembles, via Grothendieck groups and homological algebra. We demonstrate that there is a natural dictionary relating cluster-tilting subcategories and their tilting theory to A-side tropical cluster combinatorics and, dually, relating modules over $\underline{\mathcal{T}}$ to the X-side; here $\underline{\mathcal{T}}$ is the image of $\mathcal{T}$ in the triangulated stable category of $\mathcal{C}$. Moreover, the exchange matrix associated to $\mathcal{T}$ arises from a natural map $p_{\mathcal{T}}\colon\mathrm{K}_0(\operatorname{mod}\underline{\mathcal{T}})\to\mathrm{K}_0(\mathcal{T})$ closely related to taking projective resolutions. Via our approach, we categorify many key identities involving mutation, g-vectors and c-vectors, including in infinite rank cases and in the presence of loops and 2-cycles. We are also able to define A- and X-cluster characters, which yield A- and X-cluster variables when there are no loops or 2-cycles, and which enable representation-theoretic proofs of cluster-theoretical statements. Continuing with the same categorical philosophy, we give a definition of a quantum cluster category, as a cluster category together with the choice of a map closely related to the adjoint of $p_{\mathcal{T}}$. Our framework enables us to show that any Hom-finite exact cluster category admits a canonical quantum structure, generalising results of Geiß--Leclerc--Schröer. |
| title | Cluster structures via representation theory: cluster ensembles, tropical duality, cluster characters and quantisation |
| topic | Representation Theory Quantum Algebra Rings and Algebras 13F60 (Primary), 14T10, 18F30, 18N25 (Secondary) |
| url | https://arxiv.org/abs/2411.11633 |