Download Basic Noncommutative Geometry by Masoud Khalkhali PDF

By Masoud Khalkhali

This article presents an advent to noncommutative geometry and a few of its purposes. it may be used both as a textbook for a graduate path or for self-study. it is going to be important for graduate scholars and researchers in arithmetic and theoretical physics and all those people who are attracted to gaining an realizing of the topic. One function of this e-book is the wealth of examples and workouts that support the reader to navigate during the topic. whereas heritage fabric is supplied within the textual content and in different appendices, a few familiarity with easy notions of sensible research, algebraic topology, differential geometry and homological algebra at a primary 12 months graduate point is helpful.

Developed by way of Alain Connes because the overdue Seventies, noncommutative geometry has chanced on many purposes to long-standing conjectures in topology and geometry and has lately made headways in theoretical physics and quantity concept. The publication begins with an in depth description of a few of the main pertinent algebrageometry correspondences through casting geometric notions in algebraic phrases, then proceeds within the moment bankruptcy to the assumption of a noncommutative area and the way it really is developed. 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 surroundings. purposes to integrality of noncommutative topological invariants are given as well.

Two new sections were additional to this moment variation: one matters the Gauss–Bonnet theorem and the definition and computation of the scalar curvature of the curved noncommutative torus, and the second one is a short creation to Hopf cyclic cohomology. The bibliography has been prolonged and a few new examples are offered.

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Additional info for Basic Noncommutative Geometry

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Groups which sends a commutative algebra A to the group of its n-th roots of unity. X n 1/, the quotient of the polynomial algebra by the relation X n D 1. X / D 1: In general, an algebraic group, such as GLn or SLn , is an affine group scheme, represented by its coordinate ring. See [188] for a good introduction to affine group schemes. 2 (Hopf duality). H; C/ denote its linear dual. H ˝H ; Á W H ! H : It can be checked that these operations turn H into a Hopf algebra, called the dual of H . Notice that H is commutative (resp.

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.

1) in the world of affine algebraic geometry. 20) Without loss of generality we can assume that I is an ideal in F Œx1 ; : : : ; xn . A morF m is a map f W V ! W which phism between affine varieties V F n and W n m is the restriction of a polynomial map F ! F . 3 Affine varieties and finitely generated commutative reduced algebras 25 morphisms between them form a category. 19). , if n x D 0 for some n, then x D 0. Consider the category of unital finitely generated commutative and reduced algebras and unital algebra homomorphisms.

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