  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.
