Ind-cluster algebras and infinite Grassmannians
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866908374809968640 |
|---|---|
| author | Gratz, Sira Korff, Christian |
| author_facet | Gratz, Sira Korff, Christian |
| contents | A prototypical examples of a cluster algebra is the coordinate ring of a finite Grassmannian: using the Plücker embedding the cluster algebra structure allows one to move between `maximal sets' of algebraically independent Plücker coordinates via mutations. Fioresi and Hacon studied a specific colimit of the coordinate rings of finite Grassmannians and its link with the infinite Grassmannian introduced by Sato and independently by Segal and Wilson in connection with the Kadomtsev-Petiashvili (KP) hierarchy, an infinite set of nonlinear partial differential equations which possess soliton solutions. In this article we prove that this ring is a cluster algebra of infinite rank with the structure induced by the colimit construction. More generally, we prove that cluster algebras of infinite rank are precisely the ind-objects of a natural category of cluster algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_01228 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Ind-cluster algebras and infinite Grassmannians Gratz, Sira Korff, Christian Representation Theory Mathematical Physics Commutative Algebra Combinatorics Exactly Solvable and Integrable Systems Primary: 13F60, Secondary: 14M15, 37K10, 05E05 A prototypical examples of a cluster algebra is the coordinate ring of a finite Grassmannian: using the Plücker embedding the cluster algebra structure allows one to move between `maximal sets' of algebraically independent Plücker coordinates via mutations. Fioresi and Hacon studied a specific colimit of the coordinate rings of finite Grassmannians and its link with the infinite Grassmannian introduced by Sato and independently by Segal and Wilson in connection with the Kadomtsev-Petiashvili (KP) hierarchy, an infinite set of nonlinear partial differential equations which possess soliton solutions. In this article we prove that this ring is a cluster algebra of infinite rank with the structure induced by the colimit construction. More generally, we prove that cluster algebras of infinite rank are precisely the ind-objects of a natural category of cluster algebras. |
| title | Ind-cluster algebras and infinite Grassmannians |
| topic | Representation Theory Mathematical Physics Commutative Algebra Combinatorics Exactly Solvable and Integrable Systems Primary: 13F60, Secondary: 14M15, 37K10, 05E05 |
| url | https://arxiv.org/abs/2505.01228 |