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
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