By Masoud Khalkhali

"Basic Noncommutative Geometry presents an creation to noncommutative geometry and a few of its functions. The ebook can be utilized both as a textbook for a graduate path at the topic or for self-study. it is going to be necessary for graduate scholars and researchers in arithmetic and theoretical physics and all people who find themselves attracted to gaining an realizing of the topic. One characteristic of this publication is theRead more...

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**Example text**

Gh/1 defines a right action of G2 on G1 . G2 / into a left U D kG1 -module algebra. The second action turns U into a right F -comodule coalgebra. G1 / ˝ kG2 ! G1 / induced by the right action of G2 on G1 . Details of this example can be found in [128] and [55]. 9 (Connes–Moscovici Hopf algebras). A very important example for noncommutative geometry and its applications to transverse geometry and number 1 [55], [56], theory is the family of Connes–Moscovici Hopf algebras Hn for n [57]. Rn / of diffeomorphisms of Rn and can also be thought of as deformations of the Lie algebra an of formal vector fields on Rn .

H/h D 1H , which shows that a group-like element is invertible. It is easily seen that group-like elements of H form a subgroup of the multiplicative group of H . For example, for H D CG the set of grouplike elements coincides with the group G itself. A primitive element of a Hopf algebra is an element h 2 H such that h D 1 ˝ h C h ˝ 1: It is easily seen that the bracket Œx; y ´ xy yx of two primitive elements is again a primitive element. It follows that primitive elements form a Lie algebra.

This result is a special case of the Stone duality between Boolean algebras and a certain class of topological spaces [102]. 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.