Seminar talk, 12 April 2017

Speaker: Hovhannes Khudaverdian

Title: Thick morphisms and higher Koszul brackets

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

This is the joint work with T. Voronov.