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Unformatted text preview: 2 Lie groups and algebraic groups. 2.1 Basic De&nitions. In this subsection we will introduce the class of groups to be studied. We &rst recall that a Lie group is a group that is also a di/erentiable manifold and multiplication ( x;y 7&! xy ) and inverse ( x 7! x & 1 ) are C 1 maps. An algebraic group is a group that is also an algebraic variety such that multi- plication and inverse are morphisms. Before we can introduce our main characters we &rst consider GL ( n; C ) as an a¢ ne algebraic group. Here M n ( C ) denotes the space of n ¡ n matrices and GL ( n; C ) = f g 2 M n ( C ) j det( g ) 6 =) g : Now M n ( C ) is given the structure of a¢ ne space C n 2 with the coordinates x ij for X = [ x ij ] : This implies that GL ( n; C ) is Z-open and as a variety is isomorphic with the a¢ ne variety M n ( C ) f det g : This implies that O ( GL ( n; C )) = C [ x ij ; det & 1 ] : Lemma 1 If G is an algebraic group over an algebraically closed &eld, F , then every point in G is smooth. Proof. Let L g : G ! G be given by L g x = gx . Then L g is an isomorphism of G as an algebraic variety ( L & 1 g = L g & 1 ). Since isomorphisms preserve the set of smooth points we see that if x 2 G is smooth so is every element of Gx = G . Proposition 2 If G is an algebraic group over an algebraically closed &eld F then the Z-connected components Proof. Theorem 18 in section 1.2.6 implies that every element of G is con- tained in a unique irreducible component. Theorem 3 A closed subgroup of GL ( n; C ) is a Lie group. This theorem is a special case of the fact that a closed subgroup of a Lie group is a Lie group. We should also explain what ¡is£means in these contexts. The result needed is Theorem 4 Let G and H be Lie groups then a continuous homomorphism f : G ! H is C 1 . 1 This implies that there is only one Lie group structure associated with the structure of G as a topological group. If G is a closed subgroup of GL ( n; C ) then we de&ne the Lie algebra of G to be Lie ( G ) = f X 2 M n ( C ) j e tX 2 G for all t 2 R g : Some explanations are in order. First we de&ne h X;Y i = tr XY & and k X k = p h X;X i We use this to de&ne the metric topology on M n ( C ) . We note that k XY k & k X kk Y k so if we set e X = 1 X m =0 X m m ! then this series converges absolutely and uniformly in compacta. In particular the implies that X ! e X de&nes a C 1 map of M n ( C ) to GL ( n; C ) in fact real (even complex) analytic. We also note that if k X k < 1 then the series log( I ¡ X ) = 1 X m =1 X m m converges absolutely and uniformly on compacta. This says that if & > is so small that if k X k < & then & & I ¡ e X & & < 1 we have log( e X ) = log( I ¡ ( I ¡ e X )) = X . Proposition 5 Let G be a closed subgroup of GL ( n; C ) then Lie ( G ) is an R- subspace of M n ( C ) such that if X;Y 2 Lie ( G ) , XY ¡ Y X = [ X;Y ] 2 Lie ( G ) ....
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