Download Basic Noncommutative Geometry by Masoud Khalkhali PDF
By Masoud Khalkhali
This article presents an creation to noncommutative geometry and a few of its purposes. it may be used both as a textbook for a graduate direction or for self-study. it will likely be precious for graduate scholars and researchers in arithmetic and theoretical physics and all those who find themselves attracted to gaining an realizing of the topic. One function of this booklet is the wealth of examples and routines that support the reader to navigate throughout the topic. whereas historical past fabric is equipped within the textual content and in different appendices, a few familiarity with uncomplicated notions of useful research, algebraic topology, differential geometry and homological algebra at a primary 12 months graduate point is helpful.
Developed through Alain Connes because the past due Nineteen Seventies, noncommutative geometry has discovered many functions to long-standing conjectures in topology and geometry and has lately made headways in theoretical physics and quantity conception. The ebook begins with a close description of a few of the main pertinent algebrageometry correspondences via casting geometric notions in algebraic phrases, then proceeds within the moment bankruptcy to the belief of a noncommutative house and the way it really 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. purposes to integrality of noncommutative topological invariants are given as well.
Two new sections were extra to this moment version: one issues 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 awarded.
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Example text
H/ D h˝1C1˝h. A typical application of the proposition is as follows. Let H D i 0 Hi be a graded cocommutative Hopf algebra. It is easy to see that H is connected if and only if H0 D k. g/, an enveloping algebra. 4 (Compact quantum groups). 2/, for 0 < q Ä 1. , U U D U U D I . 1, but, as with compact topological groups, it has a dense subalgebra which is a Hopf algebra. 2// generated by elements ˛ and ˇ. 1 iv). 2// into a Hopf algebra as follows. Its coproduct and antipode are defined by à  à  à  ˛ ˇ ˛ ˇ .
It is obviously a unital C -algebra. a/ D haıe ; ıe i; a 2 Cr ; defines a positive and faithful trace W Cr ! C. aa / > 0 if a ¤ 0: Checking the faithfulness of the trace on the dense subalgebra C is straightforward. We refer to [72] for its faithfulness on Cr in general. The full group C -algebra of is the norm completion of C under the norm k k D supfk . /kI is a -representation of Cg; where by a -representation we mean a -representation on a Hilbert space. Note that k k is finite since for any -representation we have X X k .
The idea of symmetry in classical geometry is encoded via the action of a group or Lie algebra on a space. This can be extended to noncommutative geometry by considering an action or coaction of a Hopf algebra on a noncommutative algebra (or coalgebra) representing a noncommutative space. We look at Hopf algebra actions first. 6 (Hopf algebra actions). 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.