Download e-book for kindle: Algebraic Groups and Number Theory by Vladimir Platonov, Andrei Rapinchuk, Rachel Rowen

By Vladimir Platonov, Andrei Rapinchuk, Rachel Rowen

ISBN-10: 0125581807

ISBN-13: 9780125581806

This milestone paintings at the mathematics idea of linear algebraic teams is now to be had in English for the 1st time. Algebraic teams and quantity thought presents the 1st systematic exposition in mathematical literature of the junction of workforce idea, algebraic geometry, and quantity thought. The exposition of the subject is equipped on a synthesis of equipment from algebraic geometry, quantity thought, research, and topology, and the result's a scientific review ofalmost all the significant result of the mathematics idea of algebraic teams received up to now.

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All the eigenvalues of g, are I), and g,g, = g,g,. We call g = g,g, the Jordan decomposition. If g E G, where G c GL,(R) is an algebraic group, then g,, g, E G. Moreover, if f : G 4 H is a morphism of algebraic groups, then f (g)s = f (g,) and f (g), = f (g,). Thus we see the Jordan decomposition is independent of the matrix realization of G. Furthermore, if g E GK then g,, g, E GK (recall that K is assumed to be perfect). Analogously, any matrix X E M,(R) can be written in the form X = X, X,, where X, and X, are respectively semisimple and nilpotent matrices such that X,X, = X,X,.

Lastly, we have dim G where 1(wo) is the length of wo. = dim T 61 + [R] = dim T + 21(wo) EXAMPLE:Let G = GL,(R). Then g = M,(R). The group of all diagonal matrices is a maximal torus T of G. Write ~i for the character of T given by ~ idiag(tl,. : . , t,) H ti. Clearly, for any matrix X = ( x , ~ )E M,(R) and any t = diag(tl, . . , t,) in T we have Ad(t)(X) = (titylxij), and therefore R(T, G) = { ~ -i ~j : i # j ) . For our simple roots we can take II = { ~ i - ~ i +: l i = l , . . , n - l ) , a n d t h e n ~ ?

Then I is a finite set (say I = (1,. . , r ) ) , and G is an almost direct product of G I , . . ,G,. In particular, G is an almost direct product of simple groups. 6 (1) is called a universal covering and F = ker T the fundamental group of G. Thus, any semisimple group has a universal covering which is a direct product of simply connected simple groups. 7. A simply connected simple algebraic group is uniquely determined up to isomorphism by its root system. The root system of a simple group is irreducible and reduced, and therefore either belongs t o one of the four classical series A,, B,, C,, D,, or is one of the five exceptional systems E6, E7, E8, F4, G2.

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Algebraic Groups and Number Theory by Vladimir Platonov, Andrei Rapinchuk, Rachel Rowen


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