Abstract
We introduce a new family of Poisson vertex algebras \(\mathcal {W}(\mathfrak {a})\) analogous to the classical \(\mathcal {W}\)-algebras. The algebra \(\mathcal {W}(\mathfrak {a})\) is associated with the centralizer \(\mathfrak {a}\) of an arbitrary nilpotent element in \(\mathfrak {gl}_N\). We show that \(\mathcal {W}(\mathfrak {a})\) is an algebra of polynomials in infinitely many variables and produce its free generators in an explicit form. This implies that \(\mathcal {W}(\mathfrak {a})\) is isomorphic to the center at the critical level of the affine vertex algebra associated with \(\mathfrak {a}\).
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Acknowledgements
E.R. wishes to thank the Sydney Mathematical Research Institute and the School of Mathematics and Statistics at the University of Sydney for their warm hospitality and the SMRI International Visitor Program for the financial support.
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Communicated by Y. Kawahigashi
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Molev, A.I., Ragoucy, E. Classical \(\mathcal {W}\)-algebras for Centralizers. Commun. Math. Phys. 378, 691–703 (2020). https://doi.org/10.1007/s00220-020-03822-0
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DOI: https://doi.org/10.1007/s00220-020-03822-0