Interpolating Feigin-Frenkel Duality at the Critical Level to Matrices of Complex Size
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910222814019584 |
|---|---|
| author | Riesen, Andrew |
| author_facet | Riesen, Andrew |
| contents | In this paper, we extend Feigin-Frenkel duality at the critical level to complex rank by identifying two seemingly unrelated constructions in complex rank. On the affine side, we interpolate Molev's construction of higher Segal-Sugawara vectors and thereby describe the centers of universal affine vertex algebras at the critical level in Deligne's interpolating categories. On the $\mathcal{W}$-side, we construct the classical $\mathcal{W}$-algebras associated with Feigin's Lie algebras of complex rank $\mathfrak{gl}_λ$ and $\mathfrak{po}_λ$ as Poisson vertex algebras, realizing their Drinfeld-Sokolov reduction via an interpolated Adler-Gelfand-Dickey bracket. Upon specialization to positive integer rank in types A, B, and C, this recovers the usual Feigin-Frenkel duality at the critical level. As applications, we obtain a uniform construction of several families of higher Segal-Sugawara vectors for Lie superalgebras and recover a complex-rank analogue of the universal Bethe algebra. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_10439 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Interpolating Feigin-Frenkel Duality at the Critical Level to Matrices of Complex Size Riesen, Andrew Quantum Algebra Category Theory Representation Theory In this paper, we extend Feigin-Frenkel duality at the critical level to complex rank by identifying two seemingly unrelated constructions in complex rank. On the affine side, we interpolate Molev's construction of higher Segal-Sugawara vectors and thereby describe the centers of universal affine vertex algebras at the critical level in Deligne's interpolating categories. On the $\mathcal{W}$-side, we construct the classical $\mathcal{W}$-algebras associated with Feigin's Lie algebras of complex rank $\mathfrak{gl}_λ$ and $\mathfrak{po}_λ$ as Poisson vertex algebras, realizing their Drinfeld-Sokolov reduction via an interpolated Adler-Gelfand-Dickey bracket. Upon specialization to positive integer rank in types A, B, and C, this recovers the usual Feigin-Frenkel duality at the critical level. As applications, we obtain a uniform construction of several families of higher Segal-Sugawara vectors for Lie superalgebras and recover a complex-rank analogue of the universal Bethe algebra. |
| title | Interpolating Feigin-Frenkel Duality at the Critical Level to Matrices of Complex Size |
| topic | Quantum Algebra Category Theory Representation Theory |
| url | https://arxiv.org/abs/2505.10439 |