Session S16 - Quantum symmetries

Wednesday, July 14, 11:-1 ~ 11:24 UTC-3

The Poisson spectrum of the symmetric algebra of the Virasoro algebra

Susan Sierra

University of Edinburgh, UK   -   s.sierra@ed.ac.uk

Let $W$ be the Witt algebra of polynomial vector fields on the punctured complex plane, and let $Vir$ be the Virasoro algebra, the unique nontrivial central extension of $W$. We discuss joint work with Alexey Petukhov to analyse Poisson ideals of the symmetric algebra of $Vir$.

We focus on understanding Poisson primitive ideals, which can be given as the Poisson cores of maximal ideals of $\operatorname{Sym}(Vir)$ and of $\operatorname{Sym}(W)$. We give a complete classification of maximal ideals of $\operatorname{Sym}(W)$ which have nontrivial Poisson cores. We then lift this classification to $\operatorname{Sym}(Vir)$, and use it to show that if $\lambda \neq 0$, then $(z-\lambda)$ is a Poisson primitive ideal of $\operatorname{Sym}(Vir)$. We give applications to the Poisson geometry of the space of functions on $Vir$.

Joint work with Alexey Petukhov (Institute for Information Transmission Problems, Russia).

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