By Masoud Khalkhali

This publication presents an creation to noncommutative geometry and a few of its purposes. it may be used both as a textbook for a graduate path at the topic or for self-study. it will likely be necessary for graduate scholars and researchers in arithmetic and theoretical physics and all those people who are attracted to gaining an knowing of the topic. One characteristic of this publication is the wealth of examples and routines that support the reader to navigate during the topic. whereas historical past fabric is supplied within the textual content and in different appendices, a few familiarity with simple notions of useful research, algebraic topology, differential geometry, and homological algebra at a first-year graduate point is beneficial. constructed by way of Alain Connes because the past due Nineteen Seventies, noncommutative geometry has came across many purposes to long-standing conjectures in topology and geometry and has lately made headways in theoretical physics and quantity concept. The e-book starts off with a close description of a few of the main pertinent algebra-geometry correspondences through casting geometric notions in algebraic phrases, then proceeds to the belief of a noncommutative house and the way it truly is built. The final chapters care for 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 environment. functions to integrality of noncommutative topological invariants are given to boot.

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**Example text**

This result, unfortunately, does not indicate a way of extending the notion of a set to some kind of ‘noncommutative set’. As was mentioned before, the commutativity of a Boolean algebra is automatic and hence a naive approach to ‘quantizing set theory’ via ‘noncommutative Boolean algebras’ is doomed to fail. 7 From groups to Hopf algebras and quantum groups The game that we have been playing so far in this chapter should be familiar by now. We encode geometric or topological structures on a space in terms of a suitable algebra of functions on that space and then try to see how much of this structure makes sense without the commutativity hypothesis on the part of the algebra.

Notice that a derivation is determined by its values on a set of generators for the algebra. It is easy to see that the following formulae define derivations ı1 ; ı2 W AÂ ! 9) has a beautiful invariance property which is the noncommutative analogue of the invariance property of the Haar measure for the torus. a/ for all a 2 AÂ ; i D 1; 2. 1. Show that the map 2 X ! X /; x 7! 1 is a homeomorphism of topological spaces. 2. Show that the category of commutative C -algebras and C -morphisms is anti-equivalent to the category of pointed compact Hausdorff spaces and base point preserving continuous maps.

We look at Hopf algebra actions first. 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.