Read e-book online Algebraic Geometry IV: Linear Algebraic Groups Invariant PDF

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

ISBN-10: 3642081193

ISBN-13: 9783642081194

ISBN-10: 366203073X

ISBN-13: 9783662030738

The difficulties being solved via invariant concept are far-reaching generalizations and extensions of difficulties at the "reduction to canonical shape" of assorted is sort of an analogous factor, projective geometry. items of linear algebra or, what Invariant idea has a ISO-year historical past, which has visible alternating classes of progress and stagnation, and adjustments within the formula of difficulties, equipment of answer, and fields of program. within the final twenty years invariant conception has skilled a interval of development, influenced through a prior improvement of the idea of algebraic teams and commutative algebra. it's now seen as a department of the speculation of algebraic transformation teams (and less than 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'll additionally suppose the reader is aware the elemental suggestions 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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4). This proof uses that the root datum determines the algebra DG • Here no special information is required about the structure constants. 3. Existence. In the proof of the existence part (i) of the theorem, given in [Sp3, Ch. 12], first the Lie algebra is constructed of a suitable semi-simple group with the given root system. This proof uses little information about Lie algebras. The existence proofs are inspired by Chevalley's original construction of the "Chevalley groups" [C2], the analogues over arbitrary fields of the complex semi-simple Lie groups.

Then H is reductive if and only if any rational representation of H is fully reducible. 3, it works in arbitrary characteristic). To prove the "only if"-part one first observes (this is also elementary) that it suffices to establish: if ¢J: H -+ GL(V) is a rational representation of the reductive group H and if v E V is a non-zero fixed vector, there exists an H -stable hyperplane in V which does not contain v. 4. One establishes that if t/I is an arbitrary rational representation of H then C defines a linear map t/I(C) of the underlying vector space which commutes with t/I( G).

If S is a subtorus of G then Ru(ZG(S» c Ru(G). 5, proposition. Corollary 2. If G is reductive then ZG(T) = T. Hence Cartan subgroups and maximal tori coincide. 1 we now deduce the following properties. Proposition. Let G be reductive, let R be the root system of (G, T). (i) The roots of R are the non-zero weights of T in the Lie algebra of G; (ii) For any rJ. (t)a); (iii) If B is a Borel subgroup containing T then rJ. E R+(B) if and only if Xa c B. The Xa are the root subgroups of G associated to T.

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


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