By Masoud Khalkhali

ISBN-10: 3037190612

ISBN-13: 9783037190616

"Basic Noncommutative Geometry presents an creation to noncommutative geometry and a few of its purposes. The booklet can be utilized both as a textbook for a graduate path at the topic or for self-study. it will likely be beneficial 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 theRead more...

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2g; where 2 ´ f0; 1g. Note that B is a complete atomic Boolean algebra. Any map f W S ! g/ ´ g B f , and S Ý 2S is a contravariant functor from the category of sets to the category of complete atomic Boolean algebras. B; 2/; where we now think of 2 as a Boolean algebra with two elements. Any algebra map f W B ! C induces a set map fO W Cy ! By by fO. / D B f for all 2 Cy . 7 From groups to Hopf algebras and quantum groups 27 categories, quasi-inverse to each other. Thus once again we have a duality between a certain category of geometric objects, namely sets, and a category of commutative algebras, namely complete atomic Boolean algebras.

This result was later sharpened by Grothendieck, Deligne, and, independently, Doplicher and Roberts. Note that the dual, in any sense of the word, of a noncommutative group cannot be a group 42 1 Examples of algebra-geometry correspondences and one is naturally interested in extending the category of groups to a larger category which is closed under duality and hopefully is even equivalent to its second dual, much as is the case for locally compact abelian groups. The Hopf–von Neumann algebras of G.

7. We saw that when Â is rational AÂ (resp. AÂ ) is isomorphic to the algebra of continuous (resp. smooth) sections of a bundle of full matrix algebras on the torus T 2 . This in particular implies that AÂ (resp. T 2 / (resp. T 2 //. Morita equivalent algebras, to be defined in the next chapter, have equivalent categories of modules, projective modules, and finite projective modules. Using Swan’s theorem, this implies that, for rational Â, there is a one-to-one correspondence between (isomorphism classes of) finite projective modules on AÂ and vector bundles on T 2 .

### Basic noncommutative geometry by Masoud Khalkhali

by Brian

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