Download Algebraic Geometry IV: Linear Algebraic Groups Invariant by T. A. Springer (auth.), A. N. Parshin, I. R. Shafarevich PDF

By T. A. Springer (auth.), A. N. Parshin, I. R. Shafarevich (eds.)

The difficulties being solved through invariant concept are far-reaching generalizations and extensions of difficulties at the "reduction to canonical shape" of varied is nearly an analogous factor, projective geometry. gadgets of linear algebra or, what Invariant conception has a ISO-year background, which has visible alternating sessions of progress and stagnation, and alterations within the formula of difficulties, tools of resolution, and fields of program. within the final twenty years invariant thought has skilled a interval of development, influenced by way of a prior improvement of the speculation of algebraic teams and commutative algebra. it's now considered as a department of the idea of algebraic transformation teams (and below a broader interpretation could be pointed out with this theory). we are going to freely use the idea of algebraic teams, an exposition of which are discovered, for instance, within the first article of the current quantity. we'll additionally suppose the reader is aware the fundamental thoughts and easiest theorems of commutative algebra and algebraic geometry; whilst deeper effects are wanted, we'll cite them within the textual content or offer appropriate references.

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1. A root datum is a quadruple 'P = (X, R, Xv, RV) where X and XV are free abelian groups, in duality by a pairing Rand R v are finite subsets of X resp. Xv, such that there is a bijection CX~CXV of R onto RV. For cx E R define endomorphisms s~ of X resp. 2. The following axioms are imposed: (RD 1) If cx E R then

Xm) in (k*t then the elements of Sm permute the Xi and the elements of {I, _l}m send (Xl' ... , xm) into (X~', ... , x~",), where Ci = ± 1. (c) G = S02m+l (char(k) -# 2). 2 (c). A maximal torus Tin G is given by the following elements { t(e J = X~i' t(e i+m) = Xi-I em+i (1:::; i :::; m), t(e 2m +l) - e 2m + l · We have ZG(T) = T. The Weyl group is as in (b) and the action on T is similar. (d) G = S02m(char(k) -# 2). View S02m as a subgroup of S02m+l in an obvious way. Then the maximal torus T of S02m+l is also one for S02m' The Weyl group is now the semi-direct product of Sm and the subgroup of {I, _1}m of the elements CI ' ...

Using Borel's fixed point theorem one sees that Rad(G) is the identity component of the intersection of all Borel groups of G. 3) one concludes that in a reductive group G the radical Rad(G) is a torus in the center of the identity component GO. The structure theory of reductive groups is a central part of the theory of linear algebraic groups. It will be reviewed in the next section. § 4. Reductive Groups G denotes a connected linear algebraic group and T a maximal torus of G. 1. Groups of Rank One.

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