By Masoud Khalkhali
This article offers 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 will likely be invaluable for graduate scholars and researchers in arithmetic and theoretical physics and all people who find themselves attracted to gaining an figuring out of the topic. One function of this e-book is the wealth of examples and workouts that aid the reader to navigate in the course of the topic. whereas heritage fabric is supplied within the textual content and in numerous appendices, a few familiarity with simple notions of practical research, algebraic topology, differential geometry and homological algebra at a primary yr graduate point is helpful.
Developed through Alain Connes because the overdue Seventies, noncommutative geometry has came upon many functions to long-standing conjectures in topology and geometry and has lately made headways in theoretical physics and quantity concept. The booklet begins with a close description of a few of the main pertinent algebrageometry correspondences by way of casting geometric notions in algebraic phrases, then proceeds within the moment bankruptcy to the belief of a noncommutative house and the way it's built. 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 atmosphere. functions to integrality of noncommutative topological invariants are given as well.
Two new sections were extra to this moment version: 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 quick creation to Hopf cyclic cohomology. The bibliography has been prolonged and a few new examples are awarded.
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Extra info for Basic Noncommutative Geometry
An important idea in Hopf algebra theory is that commutative or cocommutative Hopf algebras are closely related to groups and Lie algebras. G/ above. We give a few more examples to indicate this connection. 1. Let G be a discrete group (it need not be finite) and let H D CG denote the group algebra of G. A typical element of CG is a finite formal linear combination P g2G ag g with ag 2 C. Group multiplication in G then defines the multiplication of 32 CG. g/ D 1 ; for all g 2 G, and linearly extend them to H .
Cocommutative) if and only if H is cocommutative (resp. commutative). We also note that H D H as Hopf algebras. G/. The isomorphism is Pis a finiteP induced by the map ag g 7! ag ıg . We note that the linear dual of an infinite dimensional Hopf algebra is not a Hopf algebra. The main problem is that when we dualize the product we only obtain a map m W H ! H ˝ H / . Notice that the dual of a coalgebra is always an algebra. One way to get around this problem is to consider the smaller restricted dual of Hopf algebras which are always a Hopf algebra , .
Let G be a group. Can we encode the group structure on G in terms of the algebra of functions on G? The answer is yes and by relaxing the commutativity assumption on the resulting structure we obtain an object which in many ways behaves like a group or a Lie algebra but is neither of them. It is called a Hopf algebra. Quantum groups are closely related objects. Algebraic quantum groups are, in general noncommutative and non-cocommutative, Hopf algebras. 1) (iv) for more on this). We start with a simple example.