By Michael Atiyah

ISBN-10: 0198532784

ISBN-13: 9780198532781

Professor Atiyah is among the maximum residing mathematicians and is celebrated in the course of the mathematical international. he's a recipient of the Fields Medal, the mathematical identical of the Nobel Prize, and continues to be on the top of his profession. His large variety of released papers, concentrating on the parts of algebraic geometry and topology, have right here been gathered into six volumes, divided thematically for simple reference by means of members attracted to a selected topic. Volumes III and IV conceal papers written in 1963-84 and are the results of an extended collaboration with I. M. Singer at the Index thought of elliptic operators.

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**Additional resources for Collected Works: Volume 4, Index Theory: 2**

**Sample text**

2 . This formula is familiar in form, if not in meaning. It looks just like the formula expressing the co-associativity of the co-multiplication in a Hopf algebra (or even in a co-algebra). Coincidences like this are not "accidents" in mathematics. What is happening here is that every Hopf algebra is a left co-module over itself with the co-action being the co-multiplication. One more word of caution: Each of the double indices in ! 10/ and ! 3). 2). It is ! D ".! 0/ /! 4) for all ! 2 . Again, this is reminiscent of a formula for Hopf algebras.

We will see later how this concept comes back to the commutative setting of finite groups in Chapter 13 to give us a new way of introducing infinitesimal structures there. 2, we have seen for the first time a q-deformation. These give many, though by no means all, examples of objects in noncommutative geometry. We will comment more on q-deformations in the notes at the end of Chapter 12. 1 Definitions We now assume that A is a Hopf algebra. The co-multiplication in A is denoted by W A ! A ˝ A; and the co-inverse (also known as the antipode) is denoted by Ä W A !

A for which we have the following two commutative diagrams: ˆ ! ? A ? 7) ˆ˝id ! ˝A˝A and ˆ ? A ? 8) Š ! ˝ C: One also says that a vector space together with a given right co-action ˆ is a right A-co-module. 1. 8). 2. As with almost all definitions, there are trivial examples. v/ WD 1 ˝ v for all v 2 . Prove that T is indeed a left co-action. Then define the trivial right co-action of A on . There is some other terminology that is used for left and right co-actions, and this has been known to lead to a lot of avoidable confusion.

### Collected Works: Volume 4, Index Theory: 2 by Michael Atiyah

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