4 LOUI S AUSLÄNDER
We hav e (w , 0) z(N(A)) .
Hence dimz(N(A) ) o r th e dimensio n o f th e kerne l o f A i s th e onl y
invariant, an d w e hav e establishe d th e following :
Theorem 1.1.1N( . B ) an d N( B ) ar e isomorphi c i f an d onl y i f
—-—————— j . c*
A(B ) an d A( B ) hav e kernei s o f th e sam e dimension .
Definition. I f dim V = 2n an d A(B ) i s nondegenerate , w e wil l cal l
2n+l-dimensional grou p N(B ) th e Heisenber g grou p o f dim2n+l .
The Heisenber g grou p ha s tw o presentation s tha t ar e particularl y
important i n applications . Th e firs t i s N( A ) wher e
( ; , ' )
which w e wil l cal l th e basi c presentation .
The secon d i s N(D ) = N( A +B
n
), wher e
=
In N(A
n
+B ) multiplicatio n i s give n b y
( V
t
l
) ( v
2 '
t
2
) = ( v
l
+ v
2 '
t
l
+ t
2
+ V
l
C v
2
)
where C = l
n
) . Thi s wil l b e calle d dua l pairin g presentatio n o f th e
Heisenberg grou p fo r reason s tha t w e wil l mak e apparen t afte r th e followin g
2n
remark. Le t V hav e coordinate s ( x , . . . , x , y , . . . , y ) . The n multipli -
1 n 1 n
cation i n N(D ) i ß give n b y
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