$\imath$Hall algebras and $\imath$quantum groups
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2022
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866914238288625664 |
|---|---|
| author | Lu, Ming Wang, Weiqiang |
| author_facet | Lu, Ming Wang, Weiqiang |
| contents | We survey some recent development on the theory of $\imath$Hall algebras. Starting from $\imath$quivers (aka quivers with involutions), we construct a class of 1-Gorenstein algebras called $\imath$quiver algebras, whose semi-derived Hall algebras give us $\imath$Hall algebras. We then use these $\imath$Hall algebras to realize quasi-split $\imath$quantum groups arising from quantum symmetric pairs. Relative braid group symmetries on $\imath$quantum groups are realized via reflection functors. In case of Jordan $\imath$quiver, the $\imath$Hall algebra is commutative and connections to $\imath$Hall-Littlewood symmetric functions are developed. In case of $\imath$quivers of diagonal type, our construction amounts to a reformulation of Bridgeland-Hall algebra realization of the Drinfeld double quantum groups (which in turn generalizes Ringel-Hall algebra realization of halves of quantum groups). Many rank 1 and rank 2 computations are supplied to illustrate the general constructions. We also briefly review $\imath$Hall algebras of weighted projective lines, and use them to realize Drinfeld type presentations of $\imath$quantum loop algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_12416 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | $\imath$Hall algebras and $\imath$quantum groups Lu, Ming Wang, Weiqiang Quantum Algebra Representation Theory We survey some recent development on the theory of $\imath$Hall algebras. Starting from $\imath$quivers (aka quivers with involutions), we construct a class of 1-Gorenstein algebras called $\imath$quiver algebras, whose semi-derived Hall algebras give us $\imath$Hall algebras. We then use these $\imath$Hall algebras to realize quasi-split $\imath$quantum groups arising from quantum symmetric pairs. Relative braid group symmetries on $\imath$quantum groups are realized via reflection functors. In case of Jordan $\imath$quiver, the $\imath$Hall algebra is commutative and connections to $\imath$Hall-Littlewood symmetric functions are developed. In case of $\imath$quivers of diagonal type, our construction amounts to a reformulation of Bridgeland-Hall algebra realization of the Drinfeld double quantum groups (which in turn generalizes Ringel-Hall algebra realization of halves of quantum groups). Many rank 1 and rank 2 computations are supplied to illustrate the general constructions. We also briefly review $\imath$Hall algebras of weighted projective lines, and use them to realize Drinfeld type presentations of $\imath$quantum loop algebras. |
| title | $\imath$Hall algebras and $\imath$quantum groups |
| topic | Quantum Algebra Representation Theory |
| url | https://arxiv.org/abs/2209.12416 |