Fox and Hedgehog Seminar Series - A froggy look at the Virasoro algebra

Freeman Dyson also compared frogs and birds: "Birds fly high in the air and survey broad vistas of mathematics out to the far horizon. ... Frogs live in the mud below and see only the flowers that grow nearby.” In this dichotomy I am definitely a frog. This talk will be about my favourite flower, rooted in the mud of noncommutative ring theory: the Virasoro algebra.

The Virasoro algebra is a Lie algebra which is important in conformal field theory, in string theory, and in representation theory. Looking at it via ring theory means considering Its universal enveloping algebra, the associated noncommutative ring, which has many interesting and mysterious properties. For example, it is one of the most noncommutative rings we know. As I will explain, noncommutative rings are nicer than commutative rings: so the universal enveloping algebra of the Virasoro algebra, which is extremely noncommutative, should be well-behaved. I will describe two theorems, one of which says this is true, and one of which says it is false. I will also discuss how studying the universal enveloping algebra leads us to theorems and conjectures about the Virasoro algebra itself.