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

By Vladimir Platonov, Andrei Rapinchuk, Rachel Rowen

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

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The decomposition G = HR,(G) described in the theorem is called the Levi decomposition. One can use it to reduce many problems to reductive groups. 3 is the analog of the theorem for Lie groups obtained by Levi and Maltsev (cf. Maltsev [1,2]). 10. Reductive groups. The basic properties of reductive groups are listed in the following theorem. 4. Let G be a reductive K-group. e. H n S is finite); (4) if char K = 0, then any algebraic representation f : G -+ GL, ( 0 ) is completely reducible. A deeper analysis of reductive, and especially semisimple, groups is based on the concept of a root system.

Note that not every K-morphism f : RLIK(G) 4 RLIK(H) has the form j = RLIK(f) for a suitable Lmorphism f : G 4 H. 4) it is easy to obtain the equality X ( R L I K ( G ) ) ~ = X(G)L for the groups of rational characters (cf. 61). Restriction of scalars has two noteworthy arithmetic properties. Let LIK be an extension of an algebraic number field and let v E VK. 3. The Lie algebra of an algebraic group. e. for any two points gl, g2 E G the translation map x t+ g2g;1x is a morphism of G as an algebraic variety, sending gl to g2.

N - 1. Note that most of the above statements do not carry over for positive characteristic. We shall require a technical assertion about unipotent groups. such that the factors Gi/Gi+1 are isomorphic to G, or (6,. 5) of K-subgroups such that the Gi/Gi+l are K-isomorphic to 6 , or G,, then G is said to be K-split. Indeed, this is equivalent to the existence of a maximal K-split torus T C G, and then any K-torus in G is K-split. 5) are K-isomorphic to 6,. 9. Connected groups. , maximal connected solvable subgroups).

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