4
THE SUPERALGEBR A Super[A ]
The hear t o f many o f the computation s o f invariant theor y i s the caleulatio n
of th e sig n o f certai n expressions . I f w E Div(^l) , w e defin e th e parity o f w,
denoted b y \w\, t o b e 0 o r 1 depending o n whethe r ^2
aeA
- Cont(tt;;a ) i s eve n
or odd . I n th e first case , w e sa y tha t w i s positive an d i n th e second , negative.
Arithmetical Operation s o n th e parit y symbo l wil l alway s b e carrie d ou t i n Z2 ;
for example , w e have \ww'\ = \w\ + |u/| .
If k i s an integer , w e set sign(fc ) = (—1)*.
2. Constructio n o f th e superalgebr a Super [A]. Le t A b e a signe d set .
We construc t a Z-algebra , Tens[A] , a s follows : A s a Z-module , Tenspi ] i s fre e
with basi s consistin g o f th e word s i n Div(A). Thus , i f p Tens[A] , then p ca n
be written uniquel y a s a finite sum , p = J2
ciwi
wher e Ci G Z an d w% G Div(A).
Multiplication i n Tens[^4 ] i s defined b y extendin g th e multiplieatio n i n Div(A) .
So, i f p i s a s abov e an d q = Yl^j
wj m
anothe r elemen t i n Tens[i4] , then pq =
^CidjWiWj wher e th e product W{Wj i s taken a s in Div(A). I n this way, Tens[A]
becomes a n associativ e algebr a wit h identity .
In Tensfyl] , w e define a n idea l IA (o r simpl y I ) t o b e th e idea l generate d b y
all expressions o f the followin g forms :
(11) uv sign(|u| \v\)vu fo r tt , v G Div(^t),
(12) aa, wher e a G A~,
(13) a^a^ - Ct
J

( i + i )
wher e a G A+.
The quotien t algebr a Tens[A]/I wil l be denote d b y Super[^4].
When th e signe d se t A i s proper , the n i n th e algebr a Super[A ] w e defin e a
sequence o f many Operation s p —• p^\ wher e p G Super[A], a s follows. Suc h a n
element p can b e writte n a s a finite linea r combinatio n
3
where Cj G Z an d mon(wj) i s the canonica l imag e o f Wj G Tens[A] i n Superpl] .
Whenever n o confusion i s possible, we shall write Wj in place of mon(t^), thu s
p =
Y^c3wJ-
j
We se t p(° ) = 1, th e identit y o f th e superalgebr a Supe r [^4]; p^ = p . I f i i s a
positive integer other than 1, the definition of pW is accomplished in the followin g
Steps:
Step 1. I f p = a , wit h a G A~, a set p M = 0 if % 1.
Step 2 . I f p = a W wit h aeA+, se t
p {a )
~i\(k\y
a
*
Step 3 . I f p = 2/12/2 2/n, with # G i4d, set p W = y[ %)y^%) ytf.
Step 4 . I f p = cmon(ty) , wher e c G Z an d w G Div(A), set p W = c*w^.
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