Get Algebraic Groups PDF

By Jürgen Müller

Show description

Read Online or Download Algebraic Groups PDF

Similar algebraic geometry books

An Invitation to Algebraic Geometry - download pdf or read online

It is a description of the underlying rules of algebraic geometry, a few of its very important advancements within the 20th century, and a few of the issues that occupy its practitioners this day. it truly is meant for the operating or the aspiring mathematician who's unusual with algebraic geometry yet needs to achieve an appreciation of its foundations and its objectives with not less than necessities.

New PDF release: Lectures on Algebraic Statistics (Oberwolfach Seminars)

How does an algebraic geometer learning secant types extra the certainty of speculation assessments in records? Why could a statistician engaged on issue research elevate open difficulties approximately determinantal forms? Connections of this kind are on the middle of the recent box of "algebraic statistics".

Richard V. Kadison and John Ringrose's Fundamentals of the Theory of Operator Algebras, Vol. 2: PDF

This paintings and basics of the speculation of Operator Algebras. quantity I, easy conception current an creation to useful research and the preliminary basics of $C^*$- and von Neumann algebra concept in a sort compatible for either intermediate graduate classes and self-study. The authors offer a transparent account of the introductory parts of this significant and technically tough topic.

Extra resources for Algebraic Groups

Example text

B) If V is irreducible, then any open subset ∅ = U ⊆ V is dense and irreducible. c) If V is irreducible and ϕ : V → W is a continuous map, where W is a topological space, then ϕ(V ) ⊆ W is irreducible. d) V is Hausdorff if and only if the diagonal {[v, v] ∈ V × V ; v ∈ V } ⊆ V × V is closed, where V × V is endowed with the product topology. e) If V is Noetherian, then V is Hausdorff if and only if V discrete. f ) If V = ∅ is Noetherian, then it is a finite union V = V1 ∪ · · · ∪ Vr , where the Vi ⊆ V are closed and irreducible.

Thus it suffices to show that Cµ Cλ , i. e. Cµ ⊆ Cλ , since then Cµ × {B} ⊆ Cλ × {B} = Cλ × {B} ⊆ Cλ , implying Cµ ⊆ Cλ . Hence we may assume that λ = [n − k, k] n and µ = [n − k − 1, k + 1] n, for some k ∈ {0, . . , n2 − 1}. We have λ = [1n−2k , 2k ] n and µ = [1n−2k−2 , 2k+1 ] n, which immediately implies Uµ ⊆ Uλ . Hence we get Aµ ∈ Cµ ∩ Uµ ⊆ Cµ ∩ Uλ ⊆ Cµ ∩ Cλ , where Aµ is as above, implying Cµ ⊆ Cλ . 7) Corollary. For G := SLn , where n ∈ N, we have dim(Gu ) = n(n − 1). Proof. We have λ [n] for all λ n.

Let V be an affine variety over K a) Show that dim(V ) = 0 if and only if V is a finite set. Which are the irreducible varieties amongst them? b) Let V be irreducible. Show that dim(V ) is the maximum of the lengths d ∈ N0 of chains ∅ = V0 ⊂ · · · ⊂ Vd = V of closed irreducible subsets. III Exercises and references 49 c) Let V be irreducible such that K[V ] is a factorial domain. Show that any closed subset W ⊂ V having equidimension dim(W ) = dim(V ) − 1 is of the form W = V(f ) for some f ∈ K[V ].

Download PDF sample

Algebraic Groups by Jürgen Müller


by Donald
4.5

Rated 4.77 of 5 – based on 33 votes