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G/ to G. These two Hopf algebras are dual to each other in a sense to be defined below. 2. g/ be the universal enveloping algebra of g. g/ by the two-sided ideal generated by x ˝ y y ˝ x Œx; y for all x; y 2 g. It is an associative algebra and the canonical map i W g ! g/ is universal in the sense that for any other associative algebra A, any linear map ˛ W g ! x/ uniquely factorises through i . g/ ! g/ ! g/ ! X / D X for all X 2 g. g/; ; "; S / is a cocommutative Hopf algebra. g/ is the symmetric algebra of g.

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. (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.

P; q/ 7! A/ defines a finite projective A-module and that all finite projective A-modules are obtained from an idempotent in some matrix algebra over A. A/ associated to a finite projective A-module P depends of course on the choice of the splitting P ˚ Q ' An . A/ the corresponding idempotent. An ; Am / as compositions u W Am ! P ˚ Q ! P ! P ˚ Q0 ! An ; v W An ! P ˚ Q0 ! P ! P ˚ Q ! Am : We have uv D e; vu D f: In general, two idempotents satisfying the above relations are called Murray–von Neumann equivalent.