W-algebras – introduction, screenings and quantum groups

Lecture 1 Boris FEIGIN

Course Description
Sometimes people use the words vertex operator algebra and W-algebra as synonyms. This is partly correct, but not entirely. Theory of W-algebras is a collection of extremely interesting examples of new algebraic objects and theory of vertex algebra is an attempt to understand and find some order in this zoo. Our lectures are the introduction – so we concentrate on examples and simplest methods of constructing the W-algebras. Note that W-algebras are deeply connected with 2-dimensional conformal field theory. So it is not possible to talk about W-algebras and do not mention some facts from the algebraic geometry of the curves. This short course consists of an introduction to random dynamical systems, from a predominantly geometric point of view. The aim is to introduce basic concepts in the context of simple examples. We will discuss some elementary results and highlight open questions.
1. Clifford algebra, Lattice vertex operator algebras.
2. Coinvariants and vertex operators.
3. Subalgebras in lattice vertex algebras. Screenings. Quantum groups and screenings.
4. Fermionic screenings. Algebra \( \hat{\delta I} (2) \) on a critical level.
5. Deformation of universal enveloping of the Lie algebra of differential operators on the circle.
6. Plane partitions and W-algebras. (something about recent progress)