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# Clifford Algebras and the Classical Groups by Ian R. Porteous

By Ian R. Porteous

This booklet displays the transforming into curiosity within the conception of Clifford algebras and their functions. the writer has transformed his past publication in this topic, Topological Geometry, and has accelerated and extra fabric. As within the past model, the writer comprises an exhaustive therapy of all of the generalizations of the classical teams, in addition to a very good exposition of the class of the conjugation anti-involution of the Clifford algebras and their complexifications. towards the tip of the e-book, the writer introduces rules from the idea of Lie teams and Lie algebras. This remedy of Clifford algebras might be welcomed via graduate scholars and researchers in algebra.

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A. For every pair n,n, where n is a natural number, n is a group (abelian if n > 2) there casts a CW-complex K(n,n) such that H n ( X , n ) = [X,K(n,n)] as functors on the homotopy category of CWcomplexes. b. The object K(n,n) of this category can be uniquely characterized by the following properties: n,(K(n,n)) = 0 for i # n, nn(K(n,n)) = n. 3. Torsors. If G is a non-commutative group one can define a set H1(X, G), Properties of this set constitute the most simple and useful part of the still fragmentary noncommutative cohomology theory.

14. About Proofs in Abelian Categories. The main thesis in working with abelian categories is that if a statement involving a finite number of objects and a fmite number of morphisms is true in the category of modules over a ring, the it remains true in an arbitrary abelian category. The thesis is based on the following embedding theorem. 1. Theorem. Let A be an abelian categorg whose objects form a set. Then there exists a ring R and an exact functor F : A + R-mod, which is an embedding on objects and an isomorphism on Hom's.

Torsors. If G is a non-commutative group one can define a set H1(X, G), Properties of this set constitute the most simple and useful part of the still fragmentary noncommutative cohomology theory. a. Let G be a topological group, X a topological space. A principal Gfibration, or simply G-torsor on X consists of the following data: a continuous map n : P + X; a continuous action G x P -+ P, (g,p) -+ gp, such that n(p) = n(gp) and locally n is isomorphic to a projection G x U -+ U with the fiberwise action g(h,p) = (gh,p).