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

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

The difficulties being solved through invariant thought are far-reaching generalizations and extensions of difficulties at the "reduction to canonical shape" of varied is sort of a similar factor, projective geometry. gadgets of linear algebra or, what Invariant conception has a ISO-year historical past, which has noticeable alternating sessions of progress and stagnation, and adjustments within the formula of difficulties, equipment of resolution, and fields of software. within the final twenty years invariant concept has skilled a interval of progress, prompted by means of a prior improvement of the speculation of algebraic teams and commutative algebra. it's now seen as a department of the idea of algebraic transformation teams (and below a broader interpretation may be pointed out with this theory). we'll freely use the idea of algebraic teams, an exposition of that are came upon, for instance, within the first article of the current quantity. we are going to additionally suppose the reader understands the elemental ideas 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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The dimension of a maximal torus of G is the rank of G. Let T be a maximal torus, C = ZG(T) its centralizer and N = NG(T) its normalizer. C is called a Cartan subgroup. Lemma. (i) C is a normal subgroup of N and the quotient W = NIC is finite; (ii) The identity component CO is nilpotent; (iii) Let X = {(xCO, y) E GICo X Glx- 1 yx E CO}. The image of the projection map X -+ G contains a non-empty open subset of G. 3) and is in fact true for an arbitrary subtorus of G. (ii) follows by observing that CO IT is unipotent.

Springer if ({3, a V> < (a, {3v> < 0 (resp. (a,{3v> < ({3,a v > < 0). It follows from the lemma that if f0 is given one can recover the integers (a, {3 v >, a, {3 E D. Another way of describing these integers is via a Cartan matrix. Assume given a numbering D = (a l , ... , an). j~n' where cij = (a;,a/>. Both the Dynkin diagram and the Cartan matrix determine the root system R up to isomorphism. For more details about these matters see for example [B02, Ch. VI]. Now assume G to be quasi-simple.

Both the Dynkin diagram and the Cartan matrix determine the root system R up to isomorphism. For more details about these matters see for example [B02, Ch. VI]. Now assume G to be quasi-simple. 6). This is equivalent to f0 being connected. 6 that the quasi-simple groups G are classified by connected Dynkin diagrams, together with a subgroup of a finite group A(f0) (viz. 6). It should be noticed that two distinct subgroups of A(f0) might lead to isomorphic algebraic groups, because of the presence of outer automorphisms.

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