Thick morhisms and higher Koszul brackets

Dr Hovhannes Khudaverdian (University of Manchester)

Frank Adams Room 1,

It is a classical result in Poisson geometry that the cotangent bundle of a Poisson manifold has the structure of a Lie algebroid. One of the manifestations of this structure is the "Koszul bracket" of differential forms (first discovered in integrable systems in 1970s). There is a natural homomorphism from the resulting Lie superalgebra into the superalgebra of multivector fields with respect to the canonical Schouten bracket.

In the talk, we shall present a homotopy analog of the above results.  When an ordinary Poisson structure is replaced by a homotopy one, instead of a single Koszul bracket there arises an infinite sequence of "higher Koszul brackets" introducing an $L_{\infty}$-algebra structure in the space of differential forms.  (See Khudaverdian-Voronov, 2008, http://arxiv.org/abs/0808.3406.)  We shall show how to use thick morphisms of supermanifolds to construct a non-linear transformation, which is an $L_{\infty}$-morphism, from this $L_{\infty}$-algebra of differential forms to the Lie superalgebra of multivector fields with the canonical Schouten bracket.

(A thick morphism of (super)manifolds is a new notion recently introduced, see e.g. http://arxiv.org/abs/1411.6720v5.)

(Based on joint work with Th. Voronov.)

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