## 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 conception of linear algebraic teams is now to be had in English for the 1st time. **Algebraic teams and quantity concept presents the 1st systematic exposition in mathematical literature of the junction of staff conception, algebraic geometry, and quantity thought. The exposition of the subject is outfitted on a synthesis of equipment from algebraic geometry, quantity conception, research, and topology, and the result's a scientific assessment ofalmost the entire significant result of the mathematics conception of algebraic teams received so far.
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**Example text**

Suppose a K-split torus T acts by automorphisms on a connected unipotent K-group U. Then for any T-invariant K-subgroup V c U we can find a T-invariant Zariski-closed subset P c U defined over K such that the product morphism induces K-isomorphisms of varieties P x V 2 U and V x P 7U. Moreover, if U is abelian then P can be chosen to be a suitable K-subgroup of U. Indeed, if U is abelian then the map I: U + L(U) introduced above is a group isomorphism; so it suffices to choose a T-invariant K-complement W c L(U) of L(V) and to set P = e(W).

For example, this is the case with reduction theory in Chapter 4, where one can even assume R = C. 1) G by (x, y) G x G ~ given H GIG given by x H x-I xy Chapter 2. 1. Structural properties of algebraic groups satisfying the usual group axioms. In principle, when working with groups of points over arbitrary rings sometimes a schematic approach is useful, too, however we did our best not to use it extensively. , is isomorphic to a Zariski-closed subgroup of a suitable GL,(R). ) Since no more general algebraic groups than linear ones will be considered in this book, the word "linear" will frequently be omitted.

Computing its differential at 1 and taking into account the decomposition g = L(T) @ (@,ERga), one can show that cp is dominant, from which it follows that the "large cell" is an open subset of G. , is a birational isomorphism, implying that G is a rational variety. 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,.