# Algebraic Transformation Groups: An Introduction by Hanspeter Kraft

By Hanspeter Kraft

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Intégration: Chapitres 7 et 8

Intégration, Chapitres 7 et 8Les Éléments de mathématique de Nicolas BOURBAKI ont pour objet une présentation rigoureuse, systématique et sans prérequis des mathématiques depuis leurs fondements. Ce quantity du Livre d’Intégration, sixième Livre du traité, traite de l’intégration sur les groupes localement compacts et de ses functions.

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Example text

4. 3). Since (E + εF )t J(E + εF ) = J + ε(F t J + JF ) we see that Lie Spn is a subspace of {F ∈ M2m | F t J + JF = 0}. The dimension of this space is 2m+1 , because 2 J t = −J and so the equation means that JF is symmetric. e. it is closed under the bracket [A, B] = AB − BA. The Lie algebra will also be denoted by sp2m , sp(V, β) or sp(V ). Using U V ] one finds the block form F = [ W Z Lie Sp2m = { U W dim gln = dim GLn = n2 , dim son = dim On = dim SOn = = sp(V ), V ∈ M2m | V, W symmetric}.

Hint: Use that Aut(C \ {z1 , z2 , . . 4. Connected component. Next we show that the underlying variety of an algebraic group is nonsingular. More precisely, we have the following result. 1 Prop. 1. Proposition. e. they are pairwise disjoint. In particular, G◦ , the connected component of the identity, is a normal subgroup of G which is both open and closed, the connected components of G are the cosets of G◦ , and the component group π0 (G) := G/G◦ is finite. Proof. 2). Since left multiplication by an element g ∈ G is an isomorphism, the open set gU also consists of nonsingular points, and the same holds for g∈G gU = G.

Exercise. Let G be an algebraic group and A ⊆ B ⊆ G “abstract” subgroups. If A is normal ¯ (resp. central) in B, then so is A¯ in B. Exercise. (1) The only algebraic group structure on the affine line C with identity element e = 0 is C+ . (Hint: If g ∗ h is such a multiplication, then g ∗ z = a(g)z + b(g) where a(g) ∈ C∗ and b(g) ∈ C. ) (2) The only algebraic group structure on C \ {0} with identity e = 1 is C∗ . ) (3) There is no algebraic group structure on C \ {z1 , z2 , . . , zr } for r > 1.