Abstract
The purpose of this article is to investigate relations between W-superalgebras and integrable super-Hamiltonian systems. To this end, we introduce the generalized Drinfel’d–Sokolov (D–S) reduction associated to a Lie superalgebra \({\mathfrak{g}}\) and its even nilpotent element f, and we find a new definition of the classical affine W-superalgebra \({\mathcal{W}(\mathfrak{g},f,k)}\) via the D–S reduction. This new construction allows us to find free generators of \({\mathcal{W}(\mathfrak{g},f,k)}\), as a differential superalgebra, and two independent Lie brackets on \({\mathcal{W}(\mathfrak{g},f,k)/\partial \mathcal{W}(\mathfrak{g},f,k).}\) Moreover, we describe super-Hamiltonian systems with the Poisson vertex algebras theory. A W-superalgebra with certain properties can be understood as an underlying differential superalgebra of a series of integrable super-Hamiltonian systems.
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Communicated by Y. Kawahigashi
This work was supported by NRF Grant # 2016R1C1B1010721.
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Suh, U.R. Classical Affine W-Superalgebras via Generalized Drinfeld–Sokolov Reductions and Related Integrable Systems. Commun. Math. Phys. 358, 199–236 (2018). https://doi.org/10.1007/s00220-017-3014-7
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DOI: https://doi.org/10.1007/s00220-017-3014-7