Decomposition theorem for minimal W-algebras

Juillard, Thibault (University of Hamburg)

Thursday 3rd September 15:00-16:00
Maths 311B

Abstract

Affine (respectively finite) W-algebras form a family of vertex (resp. associative) algebras which are quantisations of the natural Poisson structure on Slodowy slices. They are constructed by quantum Hamiltonian reduction. Each family is indexed by conjugacy classes of $\mathfrak{sl}_2$-triples in simple Lie algebras and minimal W-algebras correspond to these triples where the nilpotent part lies in the minimal nilpotent orbit of the simple Lie algebra.

In this talk, I will explain how minimal finite W-algebras can be reconstructed as an invariant subalgebra of a suitable localisation of a universal envelopping algebra. In fact, this localisation naturally decomposes as the tensor product of the W-algebra and another subalgebra.  The same holds in the vertex-algebraic setting. These results are motivated by finding a systematic way to construct inverse Hamiltonian reductions for minimal W- algebras, a very powerful tool in representation theory and mathematical physics.  

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