# Linear Algebraic Groups by T.A. Springer

By T.A. Springer

*"[The first] ten chapters...are an effective, obtainable, and self-contained advent to affine algebraic teams over an algebraically closed box. the writer comprises workouts and the ebook is unquestionably usable via graduate scholars as a textual content or for self-study...the writer [has a] student-friendly variety… [The following] seven chapters... might even be an excellent advent to rationality matters for algebraic teams. a couple of effects from the literature…appear for the 1st time in a text." ***–Mathematical reports **(Review of the second one Edition)

*"This booklet is a very new edition of the 1st version. the purpose of the previous ebook used to be to provide the speculation of linear algebraic teams over an algebraically closed box. interpreting that ebook, many of us entered the study box of linear algebraic teams. the current booklet has a much wider scope. Its target is to regard the speculation of linear algebraic teams over arbitrary fields. back, the writer retains the therapy of must haves self-contained. the cloth of the 1st ten chapters covers the contents of the outdated publication, however the association is a little varied and there are additions, resembling the fundamental proof approximately algebraic kinds and algebraic teams over a flooring box, in addition to an basic remedy of Tannaka's theorem. those chapters can function a textual content for an introductory path on linear algebraic teams. The final seven chapters are new. They take care of algebraic teams over arbitrary fields. many of the fabric has no longer been handled ahead of in different texts, similar to Rosenlicht's effects approximately solvable teams in bankruptcy 14, the concept of Borel and titties at the conjugacy over the floor box of maximal cut up tori in an arbitrary linear algebraic team in bankruptcy 15, and the knockers category of easy teams over a flooring box in bankruptcy 17. The publication comprises many routines and a topic index." ***–Zentralblatt Math **(Review of the second one Edition)

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**Example text**

7 the morph isms JL and i are defined by an algebra homomorphism fj. : A ---+ A ®k A (called comultiplication) and an algebra isomorphism t : A ---+ A (called antipode). Moreover, the identity element is a homomorphism e : A ---+ k. Denote by m : A ®A ---+ A the multiplication map (so m(f ® g) = fg) and let IE be the composite of e and the inclusion map k ---+ A. The group axioms are expressed by the following properties: (associativity) the homomorphisms fj. ® id and id ® fj. of A to A ® A ® A coincide; (existence of inverse) m 0 (t ® id) 0 fj.

It follows that ¢ defines an isomorphism of algebraic groups G ~ ¢(G). This proves (i). The easy proof of (ii) is omitted. 8. Lemma. Let H be a closed subgroup ofG. (g)Io(H) = Io(H)} = {g E G I p(g)Io(H) = Io(H)}. 2. It suffices to prove this for)... (g)f E Io(H). (g)f)(e) = 0 for all f E Io(H) and g E H. 9. Exercises. 5 there exists an increasing sequence of finite dimensional subspaces (Vi) of k[X] such that (a) each Vi is stable under s( G) and s defines a rational representation of G in Vi, and (b) k[X] = Ui Vi.

Verification is left to the reader. 12 is the fact that if G is a unipotent linear algebraic group and G -+ G L(V) a rational representation, there is a non-zero vector in V fixed by all of G. This fact is used to prove the following geometric result (theorem of Kostant-Rosenlicht). 5. 14. Proposition. Let G be a unipotent linear algebraic group and X an affine G-space. Then all orbits of G in X are closed. Let 0 be an orbit. Replacing X by the closure 0 we may assume that 0 is dense in X. 3 (i), 0 is also open.