Abstract

Consider the following Poisson bracket on $\mathbb{R}^\ast \times \mathbb{R}^3$ with coordinates $(u,x,y,t)$: $$ \{x,y\} = 3 u^3, \{u,x\} = t^2 u^2, \{y,u\} = 2 t u^3, \{x,t\} = 2 t^ 3u, \{y,t\} = -t^2 u^2, \{u,t\} = 0. $$ Away from $W = \{t=0\}$, this defines a symplectic form and it drops rank along W at order 4 where it defines a symplectic foliation by codimension two leaves. This is an example of a $b^4$-symplectic structure as introduced by Scott, who describes them as symplectic structures on a Lie algebroid called the $b^4$ tangent bundle. This Poisson structure is invariant under the $\mathbb{Z}^2$-action translating the $x, y$ coordinates. However, the induced Poisson structure on $\mathbb{R}^\ast \times \mathbb{R}^3/\mathbb{Z}^2 = \mathbb{R}^\ast \times T^2 \times \mathbb{R}$ will no longer be a $b^4$ symplectic structure. We introduce a class of Lie algebroids called Hypersurface (HS) algebroids, which generalize Scott's $b^k$-tangent bundles, and can describe the Poisson structure induced above. Interestingly, the Poisson bracket above is the first of an infinte family of Poisson structures arising from the group $G_k = \{a_0 z + a_1 z^2 + ... + a_{k-1} z^k\}$ of degree $k$ truncated polynomials in one variable. Even more suprising, the compact leaves of these Poisson structure have been studied before by Babenko and Taimanov who used them to contruct examples of non-formal simply connected symplectic manifolds. Joint work with Francis Bischoff.