By J.N. Coldstream

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**Example text**

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/ be the Hopf algebra of representable functions on G. There is a canonical non-degenerate pairing H ˝ K ! e tX /j tD0 dt (cf. [85] for a thorough discussion). We shall see that there is an analogous pairing between compact quantum groups of classical Lie groups and their associated quantized enveloping algebras (cf.

A morphism of complete Boolean algebras is a unital ring map which preserves all infs and sups. (Of course, any unital ring map between Boolean algebras preserves finite sups and infs). Now, given a set S let B D 2S D ff W S ! 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.

F U' / D y1n @f U ; @y ' Ã d' dn log f U' : dy n dy Once these formulas are given, it can be checked, by a long computation, that A is indeed an H1 -module algebra. In the original application, M is a transversal for a codimension one foliation and thus H1 acts via transverse differential operators [55]. Remark 1. The theory of Hopf algebras and Hopf spaces has its roots in algebraic topology and was born in the paper of H. Hopf in his celebrated computation of the rational cohomology of compact connected Lie groups [101].

### A Protogeometric Nature Goddess from Knossos by J.N. Coldstream

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