W-algebras

Lecture 1: W-algebras are vertex algebras associated to affine Lie superalgebras via quantum Hamiltonian reduction. In this first lecture I want to introduce these objects, recall its history, importance and interesting basic properties.

Lecture 2: Trialities are isomorphisms between cosets of W-superalgebras. In this lecture I want to discuss more connections between different W-superalgebras and especially how their representation categories are related.

W-algebras

Lecture 1: Orbifolds and cosets of W-algebras

For a simple Lie superalgebra g and an even nilpotent element f in g, consider the W-algebra W^k(g,f). I will discuss the following results:

(1) For any reductive group G of automorphisms of W^k(g,f), the orbifold W^k(g,f)^G is strongly finitely generated for generic values of k.

(2) For any affine vertex subalgebra V^{k'}(b) of W^k(g,f) where b is a reductive Lie algebra, the coset of W^k(g,f) by V^{k'}(b) is strongly finitely generated for generic values of k.

These are analogues of Hilbert's theorem on the finite generation of classical invariant rings. The proof involves methods of invariant theory and can be made constructive in nice enough examples. This is a key ingredient in the proof of trialities between W-algebras.

Lecture 2: Trialities of W-algebras: sketch of proof

I will outline the proof of my recent result with Thomas Creutzig which gives trialities between families of W-superalgebras of type A, as well as a similar result involving trialities in types B, C, and D.