Nonhomogeneous quadratic duality and curvature

TL;DR

This paper develops a duality theory for nonhomogeneous quadratic algebras, linking curvature, Chern classes, and classical dualities like differential operators and de Rham complexes, with implications for algebraic and geometric structures.

Contribution

It introduces quadratic duality for nonhomogeneous relations, interprets curvature via scalar relations, and discusses Chern and Chern-Simons classes within this framework.

Findings

Defined quadratic duality for nonhomogeneous algebras

Connected curvature to scalar parts of relations

Provided a simple proof of the Poincare-Birkhoff-Witt theorem

Abstract

This is a slightly corrected version of the article published by Functional Analysis and its Applications in 1993. We define the quadratic duality for algebras with nonhomogeneous relations; the duality between the algebra of differential operators and the multiplicative de Rham complex is a classical example. The scalar part of the relations is interpreted as the curvature. Chern classes of nonhomogeneneous quadratic algebras are introduced as certain obstructions; the Chern-Simons classes are also discussed. The Poincare-Birkhoff-Witt theorem is considered in the context of the duality, and its simple proof is given.