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