Nonhomogeneous quadratic duality and curvature

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.