4
1. SOME THEORY OF LINEAR ALGEBRAIC GROUPS
degree of F(K) over F. If K is not connected, then dim(if) is defined to be
dim(if°).
PROPOSITION
1.1.6. Let K be an algebraic group. Then
(a) dim(K) oo;
(b) dim(if) = 0 if and only if K is finite;
(c) If H is a closed subgroup of K, then dim(H) dim(K), with equality holding
if and only if K H;
(d) The set of connected subgroups of K satisfies the maximum condition;
(e) For any morphism \ : K H is of algebraic groups, we have dim(if) =
dim(ker(/)) -f dim(/(if)); and
(f) dim(ifi x if2) = dim(ifi) + dim(if2).
EXAMPLES
1.1.7. For any integer n 1, the following subgroups of GLn(F)
are closed: SLn(F); T
n
, the group of all upper triangular nonsingular matrices; D
n
,
the group of all diagonal nonsingular matrices; t/
n
, the subgroup of Tn consisting
of all upper triangular matrices all of whose diagonal entries equal 1.
For any n-dimensional vector space V over JP, the algebraic group structure on
GLn(F) can be transported to one on GL(V) by choice of a basis of V, and the
resulting algebraic group structure on GL(V) is independent of the choice of basis.
Stabilizers in GL(V) of subspaces, multilinear forms and quadratic forms are all
closed subgroups. Finally, the natural action of SL(V) on V*£)V yields a morphism
SL(V) - 5L(y* ® V) with kernel Z(SL(V)) and one may define PSL(V) to be
the image of this morphism.
1.2. Jordan Decomposition
DEFINITION 1.2.1. Let K be an algebraic group. A mapping tp : K K is
rational if and only if for every polynomial function / F[K], the composite foi/j
again lies in F[K]. In this case t/* : F[K] F[K] is the F-algebra homomorphism
defined by ^*(/) = / o ip.
Let g G K and let pg : K K be right translation by g (so that pg(x) = xg).
Then pg is rational, and g is called semisimple (resp. unipotent) if and only if the
jP-linear transformation p* of F[K] is semisimple (resp. locally unipotent). This
condition means that F[K] is the sum of finite-dimensional p*-invariant subspaces
on each of which the restriction of p* is diagonalizable (resp. p* 1 is nilpotent).
Finally, let ^ be an automorphism of the algebraic group K. Then ^ is
semisimple if and only if the linear transformation ip* is semisimple.
We remark that one strictly should say "g is semisimple with respect to if,"
etc., as the definition depends a priori on if. But this is unnecessary in view of
1.2.2b below.
PROPOSITION 1.2.2. Let K be an algebraic group.
(a) If (j) : if H is a morphism of algebraic groups and g G if is semisimple
{resp. unipotent), then 4(g) is semisimple (resp. unipotent);
(b) If H is a closed subgroup of K and g G H, then g is semisimple (resp.
unipotent) with respect to H if and only if it is semisimple (resp. unipotent)
with respect to K;
(c)
If 9 K
is
semisimple, then the inner automorphism ig : x i—
xg
of K is a
semisimple automorphism of K;
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