Nonhomogeneous quadratic duality and curvature

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.

arXiv commentPlain TeX, 11 pages; v.2: several notes added at the end, (Journal ref Russian)

Cited by in corpus (34)

• Loop Spaces and Connections
• Coherent analogues of matrix factorizations and relative singularity categories
• Homological algebra of semimodules and semicontramodules: Semi-infinite homological algebra of associative algebraic structures
• PBW-deformations of N-Koszul algebras
• Algebra+Homotopy=Operad
• Hochschild (co)homology of the second kind I
• Isomonodromic differential equations and differential categories
• Hilbert schemes and $y$-ification of Khovanov-Rozansky homology
• Noncommutative supergeometry, duality and deformations
• Koszulity of cohomology = $K(\pi ,1)$-ness + quasi-formality
• Compatibility with cap-products in Tsygan's formality and homological Duflo isomorphism
• Differential graded Koszul duality: an introductory survey
• Quadratic-linear duality and rational homotopy theory of chordal arrangements
• Poincare-Birkhoff-Witt Theorems
• Relative nonhomogeneous Koszul duality
• Coderived and contraderived categories of locally presentable abelian DG-categories
• Thin links and Conway spheres
• Curved infinity-algebras and their characteristic classes
• Deformations of path algebras of quivers with relations
• Deformations of vector bundles on coisotropic subvarieties via the Atiyah class
• The exterior algebra and central notions in mathematics
• Noncommutative Euclidean spaces
• Noncommutative products of Euclidean spaces
• Categories of holomorphic line bundles on higher dimensional noncommutative complex tori
• Curved Koszul duality of algebras over unital versions of binary operads
• $PBW$-deformations of graded rings
• Curved $A_{\infty }$-algebras and gauge theory
• Strictly unital A-infinity algebras
• The bar derived category of a curved dg algebra
• Generalized cohomological field theories in the higher order formalism
• Koszul duality for Iwasawa algebras modulo p
• From Koszul duality to Poincaré duality
• Contraderived categories of CDG-modules
• Curved Koszul duality and cyclic (co)homology