Graduate Student Seminar- Computing Automorphisms of Poisson Algebras

Date and Time: Friday, September 27th 12:30 - 1:30 p.m.
Place: Rome 771
Speaker: Charlene Houchins, GWU

Title: Computing Automorphisms of Poisson Algebras

Abstract: For polynomials in one or two variables over a field of characteristic zero, all automorphisms are tame. When we encounter polynomials in three variables, some automorphisms are wild. This makes it much harder to compute (tame?) the automorphisms. This means we don't explicitly know the generators of all automorphisms when we have three or more variables. However, if we introduce a Poisson bracket multiplication, there is more hope to find an explicit characterization of automorphisms. We will fix a homogeneous polynomial of the form $\Omega=x^n+y^n+z^n$ to define our Poisson bracket. We will see that for $n \geq 5$, valuation maps can be used to classify all automorphisms. Not only that, all of the automorphisms are graded. For $n=4$, we will use more elementary methods to determine the generators of our automorphism group. For $1 \leq n \leq 3$, it is unknown what the automorphisms look like, but will certainly be explored in the future.