10 S. PREVOST

Note that the degree operator d gives a natural Z-grading of g: g = IJnGZ Sfnl' w ^ e r e

g, , is the ad d eigenspace with eigenvalue n. It is well-known that g is an affine Lie

algebra of type A?\ D\l) or £,(1) (cf. [M], [F-L-M]).

The Cartan subalgebra of g is given by

hc = h e C c © a (2.1.13)

The nonsingularity of (•, •} on A x A and (2.1.12) show that the form (•, •) is nonde-

generate on the Cartan subalgebra h6- Thus we may identify h6 with its dual.

We say that a ( / 0) G h6 is a root of g if the space

g a = {x G g | [h,x] = (h,a)x for aU he he}

contains at least one non-zero element of g. By (2.1.11), (2.1.12) and (2.1.13) we see

that the set of roots of g is given by

A(g) = {ot + mc | a G A, m G Z} U {nc \ n G Z \ {0}}

(2.1.14)

= Afl(g) U A7(g).

Aj*(g) is called the set of real roots of g and A/(g) is the set of imaginary roots of g.

For a G A(g) with (a, a) ^ 0, define a v = ^ ^ . We next recall from [G] the

definition of a Chevalley basis for the affine Lie algebra g. Choose a section e : L — L.

A Chevalley basis for the affine Lie algebra g is a basis of g of the form

{xa+mc (€ g a+mc ) | a G A, n G Z} U {a v 0 t m | a € H, m € Z}

U {(«o) v | «o is the lowest root of g} U {d}

such that

1. [Za+mc, Z-a-mc] = ^(a , - a ) ( a + 77ic)V,

2. the linear map 0 : g — » g defined by

Xa+mc

H

~ ~

e

(

t t

?

~°c)x-a-mc