I. Foundations .
1.1. Bilinea r Form s an d Presentation s o f Certai n Two-Ste p Nilpoten t
Lie Groups .
Let V o r V b e a n n-dimensiona l rea l vecto r Space , an d le t
B : V X V -* IR b e a bilinea r mapping . W e begi n b y usin g B t o defin e a
nilpotent Li e grou p whic h w e wil l denot e b y N(B) . Thos e peopl e wh o ar e
familiär wit h suc h thing s wil l b e abl e t o fin d nilpoten t associativ e algebra s
lurking i n th e shadows .
Let v . V an d t . IR , i = 1,2; w e defin e a multiplicatio n o n th e se t
1 1
N(B) = V X IR b y th e formul a
1. (v
r
t
l ) (
v
2
, t
2
) = (v
1 +
v
2
, t
1 +
t
2 +
B(
V l
, v
2
)).
It i s straightforwar d t o verif y tha t N(B ) i s a grou p unde r thi s la w o f compo -
sition wit h identit y (0,0 ) V X IR an d (v,t) ~ = (-v , B(v,v)-t) .
If ( x . , . . . , x ) = v an d B = ( a . . ) , i, j = l , . . . , n, relativ e t o thi s basi s
I n i j
of V , the n i t i s eas y t o verif y tha t
((x
r
,
\ 1
is a faithfu l representatio n o f N(B) . Clearl y th e subgrou p Z = {(0 , t)| t IR }
is a norma l subgrou p o f N(B ) isomorphi c t o R an d N(B)/ Z i s isomorphi c
to V . Hence , Z D[N(B) , N(B)] , wher e [ , ] denote s th e commutato r sub -
group o f th e grou p i n th e bracket . Henc e N(B ) i s a 2-ste p nilpoten t Li e grou p
,x
n
),t )
-*• 1
/ '
1 0 .
1
. La . x . t
0 x
o :
1 X
n
http://dx.doi.org/10.1090/cbms/034/01
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