Representations of shifted quantum affine algebras and cluster algebras I. The simply-laced case
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866912092022374400 |
|---|---|
| author | Geiss, Christof Hernandez, David Leclerc, Bernard |
| author_facet | Geiss, Christof Hernandez, David Leclerc, Bernard |
| contents | We introduce a family of cluster algebras of infinite rank associated with root systems of type $A$, $D$, $E$. We show that suitable completions of these cluster algebras are isomorphic to the Grothendieck rings of the categories $\mathcal{O}_\mathbb{Z}$ of the corresponding shifted quantum affine algebras. The cluster variables of a class of distinguished initial seeds are certain formal power series defined by E. Frenkel and the second author, which satisfy a system of functional relations called $QQ$-system. We conjecture that all cluster monomials are classes of simple objects of $\mathcal{O}_\mathbb{Z}$. In the final section, we show that these cluster algebras contain infinitely many cluster subalgebras isomorphic to the coordinate ring of the open double Bruhat cell of the corresponding simple simply-connected algebraic group. This explains the similarity between $QQ$-system relations and certain generalized minor identities discovered by Fomin and Zelevinsky. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_04616 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Representations of shifted quantum affine algebras and cluster algebras I. The simply-laced case Geiss, Christof Hernandez, David Leclerc, Bernard Quantum Algebra High Energy Physics - Theory Rings and Algebras Representation Theory 17B67, 13F60, 17B10, 17B37, 82B23 We introduce a family of cluster algebras of infinite rank associated with root systems of type $A$, $D$, $E$. We show that suitable completions of these cluster algebras are isomorphic to the Grothendieck rings of the categories $\mathcal{O}_\mathbb{Z}$ of the corresponding shifted quantum affine algebras. The cluster variables of a class of distinguished initial seeds are certain formal power series defined by E. Frenkel and the second author, which satisfy a system of functional relations called $QQ$-system. We conjecture that all cluster monomials are classes of simple objects of $\mathcal{O}_\mathbb{Z}$. In the final section, we show that these cluster algebras contain infinitely many cluster subalgebras isomorphic to the coordinate ring of the open double Bruhat cell of the corresponding simple simply-connected algebraic group. This explains the similarity between $QQ$-system relations and certain generalized minor identities discovered by Fomin and Zelevinsky. |
| title | Representations of shifted quantum affine algebras and cluster algebras I. The simply-laced case |
| topic | Quantum Algebra High Energy Physics - Theory Rings and Algebras Representation Theory 17B67, 13F60, 17B10, 17B37, 82B23 |
| url | https://arxiv.org/abs/2401.04616 |