Cyclic Homology in Non-Commutative Geometry
Springer Science & Business Media, Nov 17, 2003 - Mathematics - 137 pages
Cyclic homology was introduced in the early eighties independently by Connes and Tsygan. They came from different directions. Connes wanted to associate homological invariants to K-homology classes and to describe the index pair ing with K-theory in that way, while Tsygan was motivated by algebraic K-theory and Lie algebra cohomology. At the same time Karoubi had done work on characteristic classes that led him to study related structures, without however arriving at cyclic homology properly speaking. Many of the principal properties of cyclic homology were already developed in the fundamental article of Connes and in the long paper by Feigin-Tsygan. In the sequel, cyclic homology was recognized quickly by many specialists as a new intriguing structure in homological algebra, with unusual features. In a first phase it was tried to treat this structure as well as possible within the traditional framework of homological algebra. The cyclic homology groups were computed in many examples and new important properties such as prod uct structures, excision for H-unital ideals, or connections with cyclic objects and simplicial topology, were established. An excellent account of the state of the theory after that phase is given in the book of Loday.
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A'-theory algebraic structures analytically nilpotent Banach algebras bicomplex bivariant theory bornological algebras bornology boundary operator bundle C*-algebras calculus canonical category of m-algebras Chern character classifying map cohomology compact Connes construction Cuntz cyclic cocycle cyclic complex cyclic homology cyclic theory defined definition Deformation quantization denote differential operators direct sum entire cyclic cohomology excision filtration finite Fredholm module functions functor Gerstenhaber algebra given graded algebra HCn(A Hilbert space Hochschild cochain Hochschild homology homomorphism homotopy invariance Hopf algebra idempotent index formula index theorem induces inductive system invertible element isomorphism K-theory Lie algebra linear map locally convex algebras long exact sequence manifold Math morphism multiplication Neumann algebra noncommutative geometry norm p-summable periodic cyclic homology proof Proposition pseudodifferential operators quasi-free algebras quasi-isomorphism quotient respect Rham seminorms signature operator subsets symplectic tensor algebra tensor product total complex Tsygan von Neumann algebra X-complex X(TA Z/2-graded complex