# Seminar talk, 12 April 2017

For an arbitrary manifold $\displaystyle{ M }$, we consider supermanifolds $\displaystyle{ \Pi TM }$ and $\displaystyle{ \Pi T^*M }$, where $\displaystyle{ \Pi }$ is the parity reversion functor. The space $\displaystyle{ \Pi T^*M }$ possesses canonical odd Schouten bracket and space $\displaystyle{ \Pi TM }$ possess canonical de Rham differential $\displaystyle{ d }$. An arbitrary even function $\displaystyle{ P }$ on $\displaystyle{ \Pi T^*M }$ such that $\displaystyle{ [P,P]=0 }$ induces a homotopy Poisson bracket on $\displaystyle{ M }$, a differential, $\displaystyle{ d_P }$ on $\displaystyle{ \Pi T^*M }$, and higher Koszul brackets on $\displaystyle{ \Pi TM }$. (If $\displaystyle{ P }$ is fiberwise quadratic, then we arrive at standard structures of Poisson geometry.) Using the language of $\displaystyle{ Q }$-manifolds and in particular of Lie algebroids, we study the interplay between canonical structures and structures depending on $\displaystyle{ P }$. Then using just recently invented theory of thick morphisms we construct a non-linear map between the $\displaystyle{ L_{\infty} }$ algebra of functions on $\displaystyle{ \Pi TM }$ with higher Koszul brackets and the Lie algebra of functions on $\displaystyle{ \Pi T^*M }$ with the canonical odd Schouten bracket.