Algebraic Transformation Groups: An Introduction by Hanspeter Kraft

By Hanspeter Kraft

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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.

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