2
0. Review of Semisimple Lie Algebras
Denote by $ C fj* the root system of g relative to I) (which by conven-
tion does not contain 0), whose abstract properties will be recalled in the
following section. To each root a G & corresponds a nonzero 1-dimensional
subspace of 0 called a root space:
Qa = {x Q | [fr#] = OL{K)X for all /i G f)}.
Usually we fix a simple system A C $ having £ elements and correspond-
ing positive system E
+
C $ having m elements. This defines a Cartan
decomposition g = n~ © f ) © n, where n := 0
a o
0
a
and n~ := 0
a o
0a-
The corresponding standard Borel subalgebra is b := f ) ffin, whose oppo-
site Borel subalgebra is b~ := f) © n~. These are maximal solvable subalge-
bras of 0; more generally, any such subalgebra is called a Borel subalgebra.
Any subalgebra p containing a Borel subalgebra is called parabolic] those
containing b are standard. There are 2i of these, in natural bijection with
subsets I C A (where we interpret I as the Greek letter "iota" for notational
consistency). A parabolic subalgebra decomposes as p I ffi u, where the
Levi factor I is reductive (the direct sum of its semisimple derived algebra
and an abelian subalgebra) while u is the largest nilpotent ideal of p. When
p = pi D b, the center of l\ lies in f ) and Ui C n; here the root system
$i = $ fi ZI of (the derived algebra of) li lies in $ and has I as a simple
system, whereas ui = 0 0
a
with a running over those a G
£+
not in $j.
The Lie algebra 0 acts on itself by derivations adx, where (ad#)(y) :=
[xy]. Let G be the adjoint group, generated by all automorphisms exp(adx)
with x G 0 nilpotent. All Cartan subalgebras of 0 are conjugate under
G; their common dimension l is the rank of 0. The adjoint representation
ad : 0 » Der0 is a first example of a representation of 0. The associ-
ated Killing form (x,y) :— Tr(adxad?/) is nondegenerate. The algebra
0 decomposes uniquely (up to order of summands) into the direct sum of
simple ideals. In turn, the simple Lie algebras are uniquely determined
by their (irreducible) root systems; these are classified explicitly as types
A^, B^, Q , D^, Ee, E7, Eg, F4, G2. This often makes case-by-case proofs pos-
sible.
Subalgebras of type Ai in 0 play a special role. Such an algebra is
isomorphic to sl(2, C), which has a basis (h, x, y):
*
: =
(o -1)'
x : =
( o J)'
y:=(^i
o)-
Each a G $
+
determines such a subalgebra (call it $a) with basis denoted
(A
a
, Xaj ya)-
We always work with a standard basis of 0, consisting of root vectors
Xa £ Qa and ya G 0_
a
(for a 0) together with ha [xaya] for a G A
so that all a(ha) = 2. There are many ways to choose such a basis. For
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