8 S. PREVOST
for a, b L. Assume that co(a,5) = (5,5) mod 2Z. Let e : L —• Z be a section with
corresponding 2-cocycle e0 : L x L —• Z/2Z such that
e(a) •-» ea
e(0) i-
e
0
(2.1.4)
K«o(«,/3)
for a, /? A. Let A denote the pullback of the root system A in L; thus
A = {a e L | a A}. (2.1.5)
Note that in terms of a section,
A = {e
a
, Kea | a A}. (2.1.6)
Form the vector space h. = l 0 z C , and extend (•, •) to h. x h. in the natural way.
Set £ = h.0 S3a€A ^ z o - (I n terms of a section, g = h.0 £*€A
a
, where xa = xea.)
The vector xa is to be a nonzero element subject only to the relation x^ = xa.
Define an alternating bilinear map [•, •] on j x j as follows:
[h,h] = 0
[h,a] = (h,a)xa
[ a if ab = l (2.1.7)
[xa, zb] = xab if ab e A
( 0 if a6£AU{l,Ac} ,
where fe€h and a, 6 A. (In terms of a section, we have
[h,h] = 0
[h,xa] = (h,a)xa
' 6(a,/?)a i f a + /9 = O,((a,0) = - 2 ) , (2.1.8)
[xa,xfi] = { e(a,p)xa+p i f a + /?€ A, ((a,/3) = - 1 ) ,
0 if a + /? g A U 0,
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