Yuly Billig (Carleton University)
Friday 2nd June, 12.05-12.55pm, Carslaw 159
Solving non-commutative differential equations in vertex algebras
In this talk we will show how the vertex algebra technique can be used to study representations for the Lie algebra W2 of vector fields on a 2-dimensional torus.
In a famous 1978 paper Lepowsky and Wilson showed that a module for an infinite-dimensional Heisenberg algebra admits the action of a much larger affine Kac-Moody algebra. It turns out that for W2 we have an analogous situation: W2 has a loop algebra sl2 as a subalgebra, and for a family of irreducible representations of sl2 the action can be extended to all of W2. This is done by solving certain operator-valued differential equations that arise from a vertex algebra associated with W2. Unlike the case of the Heisenberg algebra, the positive/negative parts of the loop algebra are non-commutative, which makes solving these equations rather challenging.
This is a joint work with Alex Molev and Ruibin Zhang.