Structure of classical (finite and affine) W-algebras

Abstract

First, we derive an explicit formula for the Poisson bracket of the classical finite W-algebra W^{fin}(g,f), the algebra of polynomial functions on the Slodowy slice associated to a simple Lie algebra g and its nilpotent element f. On the other hand, we produce an explicit set of generators and we derive an explicit formula for the Poisson vertex algebra structure of the classical affine W-algebra W(g,f). As an immediate consequence, we obtain a Poisson algebra isomorphism between W^{fin}(g,f) and the Zhu algebra of W(g,f). We also study the generalized Miura map for classical W-algebras.

4 pages

Cited by in corpus (13)

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• Classical affine W-algebras and the associated integrable Hamiltonian hierarchies for classical Lie algebras
• Affine Super Yangians and Rectangular $W$-superalgebras
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• Screening operators and Parabolic inductions for Affine W-algebras (with an appendix by Shigenori Nakatsuka)
• p-reduced multicomponent KP hierarchy and classical W-algebras W(gl_N,p)
• Finite W-algebras for gl_N
• Classical affine W-superalgebras via generalized Drinfeld-Sokolov reductions and related integrable systems
• On Miura maps for W-superalgebras
• Reduction by stages for finite W-algebras
• MasterPVA and WAlg: Mathematica packages for Poisson vertex algebras and classical affine $\mathcal{W}$-algebras
• Generators of Supersymmetric Classical $W$-algebras
• Universal filtered quantizations of nilpotent Slodowy slices