A decomposition theorem for minimal W-algebras of type A

arXiv:2611.13239v1 Announce Type: new Abstract: To any simple Lie algebra, one can associate a minimal affine W-algebra. It is a vertex algebra which is obtained from the universal affine vertex algebra by a chiral Hamiltonian reduction procedure. This reduction is determined by data coming from the minimal nilpotent orbit. It is a chiral quantisation of the Slodowy slice associated to this minimal orbit. In this paper, when the simple Lie algebra is of type A, we prove that the minimal W-algebra is isomorphic to a commutant subalgebra in a localisation of the universal affine vertex algebra. Then, we establish a decomposition theorem \`a la Losev: the localised universal affine vertex algebra decomposes as the tensor product of the minimal affine W-algebra and another vertex subalgebra. As a direct corollary, we retrieve the celebrated result of inverse Hamiltonian reduction for the minimal W-algebra. We also prove an analogue of this result for the corresponding finite W-algebra. In both cases, our approach relies on the geometry of the associated Slodowy slice.

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