By Larry Smith
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Extra resources for Algebraic topology: Proc. conf. Goettingen 1984
A Boolean algebra is necessarily commutative as can be easily shown. One defines an order relation on B by declaring x Ä y if there is an y 0 such that x D yy 0 . It can be checked that this is in fact a partial order relation on B. An atom in a Boolean algebra is an element x such that there is no y with 0 < y < x. A Boolean algebra is atomic if every element x is the supremum of all the atoms smaller than x. A Boolean algebra is complete if every subset has a supremum and infimum. A morphism of complete Boolean algebras is a unital ring map which preserves all infs and sups.
Let q be a nonzero complex number and let A D Cq Œx; y be the algebra of coordinates on the quantum q-plane. yx qxy/ is the two-sided ideal generated by yx qxy. For q ¤ 1, the algebra Cq Œx; y is noncommutative. 2/-comodule algebra structure W A ! y/ for the quantum plane, which is straightforward. We can also define the coaction of a Hopf algebra on a coalgebra, and the concept of a comodule coalgebra. This plays an important role in the next example. 8 (Bicrossed products). The examples of Hopf algebras that are really difficult to construct are the noncommutative and the non-cocommutative ones.
The Hopf line bundle on the two-sphere S 2 , also known as the magnetic monopole bundle, can be defined in various ways. ) Here is an approach that lends itself to noncommutative generalizations. C/ that satisfy the canonical anticommutation relations: i j C j i D 2ıij for all i; j D 1; 2; 3. Here ıij is the Kronecker symbol. A canonical choice is the so called Pauli spin matrices Â Ã Â Ã Â Ã 0 1 0 i 1 0 ; ; : 1 D 2 D 3 D 1 0 i 0 0 1 Define a function F W S 2 ! x1 ; x2 ; x3 / D x1 1 C x2 2 C x3 3; where x1 , x2 , x3 are coordinate functions on S 2 , so that x12 C x22 C x32 D 1.