Universal quantisations of nilpotent Slodowy varieties and W-algebras
Lewis Topley (University of Birmingham)
Wednesday 22nd January, 2020 16:00-17:00 Maths 311B
Namikawa proved that every conic symplectic singularity admits a universal (global) filtered Poisson deformation. This work was complemented by Losev, who demonstrated that these varieties also admit universal filtered quantisations. I will begin this talk by reviewing these definitions and recalling that these universal objects may be viewed as initial objects in appropriately defined categories. Our first result gives a simplified description of these two categories, relating them to easily-understood categories of commutative algebras.
Next we apply these results to a nilpotent Slodowy variety X, obtained from the nilpotent cone of a complex semisimple Lie algebra by Hamiltonian reduction. The cases where the Slodowy slice is the universal Poisson deformation of X are classified by Lehn--Namikawa--Sorger and we use our first result to show that the LNS theorem holds for filtered quantisations of X, using the quantum W-algebra in place of the Slodowy slice. The slice to the subregular singularity in non-simply laced types is of special interest, since these are non-universal Poisson deformations. In this case we show that the (quantum and classical) W-algebra arises as quotients of another W-algebra corresponding to an appropriate simply-laced subregular Slodowy variety. Using this we discover new Yangian-type presentations of subregular quantum W-algebras in type B.
This is a work in progress with Ambrosio--Carnovale--Esposito. I will try to keep prerequisites to a minimum.