Yuly Billig (Carleton University)
Friday 2nd June, 12.05-12.55pm, Carslaw 159Solving non-commutative differential equations in vertex algebrasIn this talk we will show how the vertex algebra technique can be used to study representations for the Lie algebra W_{2} 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 W_{2} we have an analogous situation: W_{2} has a loop algebra sl_{2} as a subalgebra, and for a family of irreducible representations of sl_{2} the action can be extended to all of W_{2}. This is done by solving certain operator-valued differential equations that arise from a vertex algebra associated with W_{2}. 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. |