By Masoud Khalkhali
"Basic Noncommutative Geometry offers an advent to noncommutative geometry and a few of its functions. The booklet can be utilized both as a textbook for a graduate direction at the topic or for self-study. it will likely be worthy for graduate scholars and researchers in arithmetic and theoretical physics and all people who find themselves drawn to gaining an figuring out of the topic. One characteristic of this ebook is the wealth of examples and workouts that aid the reader to navigate throughout the topic. whereas historical past fabric is supplied within the textual content and in different appendices, a few familiarity with easy notions of practical research, algebraic topology, differential geometry and homological algebra at a primary 12 months graduate point is beneficial. constructed through Alain Connes because the past due Nineteen Seventies, noncommutative geometry has came across many functions to long-standing conjectures in topology and geometry and has lately made headways in theoretical physics and quantity idea. The e-book starts off with an in depth description of a few of the main pertinent algebra-geometry correspondences by means of casting geometric notions in algebraic phrases, then proceeds within the moment bankruptcy to the assumption of a noncommutative house and the way it truly is developed. The final chapters take care of homological instruments: cyclic cohomology and Connes-Chern characters in K-theory and K-homology, culminating in a single commutative diagram expressing the equality of topological and analytic index in a noncommutative surroundings. functions to integrality of noncommutative topological invariants are given as well."--Publisher's description. Read more...
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Additional info for Basic noncommutative geometry
Let H be a Hopf algebra. An algebra A is called a left H -module algebra if A is a left H -module via a map W H ˝ A ! h/1 for all h 2 H and a; b 2 A. Using the relations h D h˝h and h D 1˝hCh˝1, for grouplike and primitive elements, it is easily seen that, in an H -module algebra, group-like elements act as unit preserving algebra automorphisms while primitive elements act as derivations. In particular, for H D CG the group algebra of a discrete group, an H -module algebra structure on A is simply an action of G by unit preserving algebra automorphisms of A.
One way to get around this problem is to consider the smaller restricted dual of Hopf algebras which are always a Hopf algebra , . The main idea is to consider, instead of all linear functionals on H , only the continuous ones (with respect to the linearly compact topology on H ). The restricted dual H B may be too small though. A better way to think about Hopf duality which covers the infinite dimensional case as well is via a Hopf pairing. A Hopf pairing between Hopf algebras K and H is a bilinear map h ; iW H ˝ K !
G/ ! G/ and Á W C ! G/. I ˝ / W H ! I ˝ "/ D I W H ! I ˝ S/ D Á" W H ! 22) where I denotes the identity map. G/ is a commutative Hopf algebra. h/ D h ˝ h; h ¤ 0: The general definition of a Hopf algebra is as follows. Let H be a unital algebra and let m W H ˝ H ! H and Á W C ! H denote its multiplication and unit maps, respectively. 1. H; m; Á/ endowed with unital algebra homomorphisms W H ! H ˝ H , " W H ! C and a linear map S W H ! 22) is called a Hopf algebra. We call the comultiplication, " the counit, and S the antipode of H .