## Elliptic deformation of $\mathcal{W}_N$-algebras

Jean Avan, Luc Frappat, Eric Ragoucy

SciPost Phys. 6, 054 (2019) · published 7 May 2019

- doi: 10.21468/SciPostPhys.6.5.054
- Submissions/Reports

### Abstract

We construct $q$-deformations of quantum $\mathcal{W}_N$ algebras with elliptic structure functions. Their spin $k+1$ generators are built from $2k$ products of the Lax matrix generators of ${\mathcal{A}_{q,p}(\widehat{gl}(N)_c)}$). The closure of the algebras is insured by a critical surface condition relating the parameters $p,q$ and the central charge $c$. Further abelianity conditions are determined, either as $c=-N$ or as a second condition on $p,q,c$. When abelianity is achieved, a Poisson bracket can be defined, that we determine explicitly. One connects these structures with previously built classical $q$-deformed $\mathcal{W}_N$ algebras and quantum $\mathcal{W}_{q,p}(\mathfrak{sl}_N)$.

### Cited by 2

### Ontology / Topics

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^{1}Jean Avan, -
^{2}Luc Frappat, -
^{2}Eric Ragoucy