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

The difficulties being solved by way of invariant thought are far-reaching generalizations and extensions of difficulties at the "reduction to canonical shape" of assorted is sort of an analogous factor, projective geometry. gadgets of linear algebra or, what Invariant idea has a ISO-year historical past, which has noticeable alternating classes of development and stagnation, and alterations within the formula of difficulties, equipment of resolution, and fields of program. within the final 20 years invariant thought has skilled a interval of progress, inspired through a prior improvement of the speculation of algebraic teams and commutative algebra. it truly is now considered as a department of the idea of algebraic transformation teams (and below a broader interpretation will be pointed out with this theory). we'll freely use the speculation of algebraic teams, an exposition of which might be came across, for instance, within the first article of the current quantity. we are going to additionally imagine the reader knows the fundamental thoughts and easiest theorems of commutative algebra and algebraic geometry; whilst deeper effects are wanted, we are going to cite them within the textual content or offer appropriate references.

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Unipotent), normal subgroup of G. G is semi-simple (resp. reductive) when Rad(G) (resp. Ru(G)) is trivial. It is easy to see that radicals exist. Clearly Ru(G) c Rad(G). A semi-simple group is reductive. 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.

2]. Theorem. Let G be a connected, semi-simple linear algebraic group of rank one. Then G is isomorphic to either SL 2 or PGL 2 • Here PGL 2 is the quotient of GL 2 by the one-dimensional torus of scalar multiplications. 13]). One first shows that the smooth projective curve GIB must be isomorphic to ]pl, as it has an infinite group of automorphisms fixing a point. 2. Roots. Now let G and T be arbitrary. We denote by X and XV the character (resp. 1). 4). For IX E R let 7;. be the identity component of the kernel of the character IX.

3(a». 4) where VA = {v E VI¢J(t)v = ),(t)v for all t E T}. Lemma. Assume ¢J to be irreducible. (i) There is a unique line L in V fixed by ¢J(G). It is a weight space VI'; (ii) If Jl. , O(V ~ 0 for all 0( E R+; (iii) If the weight space VA is non-zero then Jl. - A. is a sum of positive roots. > I. l) a fixed line exists for any rational representation ifJ. The essential point of (i) is the uniqueness for ifJ irreducible. 1 E X with the property of (ii) is said to be dominant. This dominant weight is the highest weight of the irreducible representation ifJ.