Seminar talk, 12 April 2017

From Geometry of Differential Equations
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Speaker: Hovhannes Khudaverdian

Title: Thick morphisms and higher Koszul brackets

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

This is the joint work with T. Voronov.