By Masoud Khalkhali

"Basic Noncommutative Geometry offers an advent to noncommutative geometry and a few of its functions. The booklet can be utilized both as a textbook for a graduate direction at the topic or for self-study. it will likely be worthy for graduate scholars and researchers in arithmetic and theoretical physics and all people who find themselves drawn to gaining an figuring out of the topic. One characteristic of this ebook is theRead more...

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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. Using the relations h D h˝h and h D 1˝hCh˝1, for grouplike and primitive elements, it is easily seen that, in an H -module algebra, group-like elements act as unit preserving algebra automorphisms while primitive elements act as derivations. In particular, for H D CG the group algebra of a discrete group, an H -module algebra structure on A is simply an action of G by unit preserving algebra automorphisms of A.

One way to get around this problem is to consider the smaller restricted dual of Hopf algebras which are always a Hopf algebra [65], [167]. The main idea is to consider, instead of all linear functionals on H , only the continuous ones (with respect to the linearly compact topology on H ). The restricted dual H B may be too small though. A better way to think about Hopf duality which covers the infinite dimensional case as well is via a Hopf pairing. A Hopf pairing between Hopf algebras K and H is a bilinear map h ; iW H ˝ K !

G/ ! G/ and Á W C ! G/. I ˝ / W H ! I ˝ "/ D I W H ! I ˝ S/ D Á" W H ! 22) where I denotes the identity map. G/ is a commutative Hopf algebra. h/ D h ˝ h; h ¤ 0: The general definition of a Hopf algebra is as follows. Let H be a unital algebra and let m W H ˝ H ! H and Á W C ! H denote its multiplication and unit maps, respectively. 1. H; m; Á/ endowed with unital algebra homomorphisms W H ! H ˝ H , " W H ! C and a linear map S W H ! 22) is called a Hopf algebra. We call the comultiplication, " the counit, and S the antipode of H .