Deformed W-algebras and chiralized cluster seeds: subregular W-algebras and Inverse Quantum Hamiltonian Reduction
Abstract
The recently introduced formalism of chiral cluster seeds replaces quantum cluster variables with deformed vertex operators. In this framework, a decorated quiver associated with a seed encodes the operator product expansions of the corresponding vertex operators. This formalism is applied to several (q,t)-deformed W-algebras, including 𝒲q,t(gl(N|M)), Uq(sl̂2), and the deformed Bershadsky--Polyakov algebra. In particular, it is shown that different free field realizations of the currents are related by mutations of the associated chiral cluster seed. The second part of the paper introduces a (q,t)-deformation of the subregular W-algebras, denoted by 𝒲q,tsub(sl(N)). All free field realizations obtainable through seed mutations are described. An embedding of 𝒲q,tsub(sl(N)) into the free field realization of 𝒲q,t(sl(N)) tensored with a rank-two Heisenberg algebra is constructed. This embedding may be viewed as a deformed analogue of inverse quantum Hamiltonian reduction. The relation between the subregular algebras and 𝒲q,t(gl(1|N)) is also discussed.
Related
- Superintegrability of discrete-time rational Ruijsenaars-Schneider model and deformed polynomial symmetry algebras
- Mirror functor for deformed preprojective algebras
- Superintegrable systems in curved spaces via coalgebras and deformed Racah algebras
- Local approximations of global Hamiltonian from inclusion of algebras
- Crepant partial resolutions of the nilpotent cone via Hamiltonian reduction
BibTeX
@misc{bershtein2026deformed,
title = {{Deformed W-algebras and chiralized cluster seeds: subregular W-algebras and Inverse Quantum Hamiltonian Reduction}},
author = {Mikhail Bershtein and Jean-Emile Bourgine and Ethan Fursman},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.30032},
eprint = {2606.30032},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.30032},
}
RIS
TY - UNPB
TI - Deformed W-algebras and chiralized cluster seeds: subregular W-algebras and Inverse Quantum Hamiltonian Reduction
AU - Mikhail Bershtein
AU - Jean-Emile Bourgine
AU - Ethan Fursman
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2606.30032
UR - https://arxiv.org/abs/2606.30032
AB - The recently introduced formalism of chiral cluster seeds replaces quantum cluster variables with deformed vertex operators. In this framework, a decorated quiver associated with a seed encodes the operator product expansions of the corresponding vertex operators. This formalism is applied to several (q,t)-deformed W-algebras, including 𝒲_q,t(gl(N|M)), U_q(sl̂_2), and the deformed Bershadsky--Polyakov algebra. In particular, it is shown that different free field realizations of the currents are related by mutations of the associated chiral cluster seed. The second part of the paper introduces a (q,t)-deformation of the subregular W-algebras, denoted by 𝒲_q,t^sub(sl(N)). All free field realizations obtainable through seed mutations are described. An embedding of 𝒲_q,t^sub(sl(N)) into the free field realization of 𝒲_q,t(sl(N)) tensored with a rank-two Heisenberg algebra is constructed. This embedding may be viewed as a deformed analogue of inverse quantum Hamiltonian reduction. The relation between the subregular algebras and 𝒲_q,t(gl(1|N)) is also discussed.
ER -