Polyvector fields and polydifferential operators associated with Lie pairs

Graph complexes in deformation quantization

Kontsevich’s formality theorem and the consequent star-product formula rely on the construction of an L∞-morphism between the DGLA of polyvector fields and the DGLA of polydifferential operators. This construction uses a version of graphical calculus. In this article we present the details of this graphical calculus with emphasis on its algebraic features. It is a morphism of differential grade...

Designing N-PolyVector Fields with Complex Polynomials

We introduce N-PolyVector fields, a generalization of N-RoSy fields for which the vectors are neither necessarily orthogonal nor rotationally symmetric. We formally define a novel representation for N-PolyVectors as the root sets of complex polynomials and analyze their topological and geometric properties. A smooth N-PolyVector field can be efficiently generated by solving a sparse linear syst...

Universal algebraic structures on polyvector fields

The theory of operads is a conceptual framework that has become a kind of universal language, relating branches of topology and algebra. This thesis uses the operadic framework to study the derived algebraic properties of polyvector fields on manifolds. The thesis is divided into eight chapters. The first is an introduction to the thesis and the research field to which it belongs, while the sec...

An L∞ algebra structure on polyvector fields

In this paper we construct an L∞ structure on polyvector fields on a vector space V over C where V may be infinite-dimensional. We prove that the constructed L∞ algebra of polyvector fields is L∞ equivalent to the Hochschild complex of polynomial functions on V , even in the infinite-dimensional case. For a finite-dimensional space V , our L∞ algebra is equivalent to the classical Schouten-Nije...

On Gelfand pairs associated with nilpotent Lie groups