Title: Structure, cohomology and deformations of local homogeneous Poisson brackets of arbitrary degree
Speaker: Guido Carlet, Institut de Mathématiques de Bourgogne, France
Time: 2025-04-15 (Tuesday) 15:20-16:20
Location: Room 1418, Management and Science Building

Abstract:
Dubrovin and Novikov initiated the study of local homogeneous differential-geometric Poisson brackets of arbitrary degree $\mathrm{<span\; style="font-size:14px;font-family:微软雅黑,\; "microsoft\; yahei";">k</span>}$ in their seminal 1984 paper. Despite many efforts, and several results in low degree, very little is known about their structure for arbitrary $\mathrm{<span\; style="font-size:14px;font-family:微软雅黑,\; "microsoft\; yahei";">k</span>}$.

After an introduction to the topic, we first report on our recent results on the structure of DN brackets of degree $\mathrm{<span\; style="font-size:14px;font-family:微软雅黑,\; "microsoft\; yahei";">k</span>}$. By applying homological algebra methods to the computation of their Poisson cohomology (or rather of an associated differential complex) we show that certain linear combinations of the coefficients of a degree $\mathrm{<span\; style="font-size:14px;font-family:微软雅黑,\; "microsoft\; yahei";">k</span>}$ DN bracket define $\mathrm{<span\; style="font-size:14px;font-family:微软雅黑,\; "microsoft\; yahei";">k</span>}$ flat connections. Moreover the Poisson cohomology of such brackets is related with the Chevalley-Eilenberg cohomology of an associated finite-dimensional Lie algebra.

In collaboration with M. Casati.