THE SUPERALGEBR A Super[A ]
5
Step 5 . I f p = ]C?=i c jwj = X^L i Pji wher e pj = CjW
3
, the n se t
pM=£pMp(-...p(H
where the su m ranges over all n-tuples o f nonnegative integer s ii, z2 ,..., i
n
suc h
that i\ 4 - «2 H h *n = i -
When A i s a signe d set , th e algebr a Super [A] wil l b e calle d th e superalgebra
generated b y th e signed se t A. Whe n A i s a proper signed set , th e superalgebr a
Super[;4] wil l b e tacitl y assume d t o b e endowe d wit h th e Operatio n p pW,
where i t wil l be calle d th e divided power Operation.
Observe th e followin g propertie s o f Super [A].
(1) W e can se t Cont(mon(w) ) = Cont(w ) an d Length(mon(w) ) = Length(w )
for w E Div[i4] , a s wel l a s |mon(it;) | = \w\. Thus , fixing a multise t m o n A,
the se t o f al l mon(ix; ) fo r w Div[^4 ] with Cont(ti/ ) = m span s a submodul e o f
Super[i4], which w e call th e submodul e o f all monomials o f content m.
Indeed, le t m b e a multiset o n A. W e consider al l words w Div(;4 ) s o tha t
Cont(w) = m . Thes e words , togethe r wit h 0 , spa n a submodul e Tens m[i4] o f
Tens[yl]. Furthermore , Tens[yl ] = ^2
m
Tens m[^4] where th e su m i s a direct sum .
Now, let i G I. The n i = X}t
m
wher e i
m
Tens
m
[j4]. Eac h i
m
i s in I sinc e thi s
property hold s fo r th e generator s o f I . I t follow s tha t i f w, wf Div(i4 ) wit h
w = w' (mo d J) bu t w & /, the n Cont(w ) = Cont(w') , Length w = Lengthw' ,
and \w\ = \w'\. W e sometimes spea k o f the "parit y o f a word i n Super[;4]."
(2) If a, b Ad, the n ab = ba in Super[yl] unless a and b are negative in which
case ab = —ba. (Thi s follow s fro m (II). )
Examples. (1) I f A = , the n Super[A ] = Symm(A) , th e Z-algebr a eon -
sisting o f al l polynomial s wit h 'Variables " th e element s a A, ofte n calle d th e
Symmetrie algebr a generate d b y th e se t A.
(2) I f A = A~, the n Super[^4 ] = Wedge(^t) , th e exterio r algebr a generate d
by th e se t A
1
tha t is , the Z-algebr a consistin g o f all skew-symmetrie monomial s
in th e 'Variables'
5
A. Thus , a monomia l i n Super[A] , sa y aia2...eifc , satisfie s
the relatio n a\a^ flfc = sgn((j)o f7ia(T2 -aak wher e o i s an y permutatio n o f
{1,2,...,*;}.
(3) If A = J 4 + , then Super[A] i s the divided powers algebra Divp[A] generate d
by th e se t A. W e recall that , i n characteristi c zer o (tha t is , over th e field Q o f
rational numbers) , th e divide d powe r a W ca n b e identifie d wit h a %/%\.
We omit th e proo f o f the followin g tw o elementary facts .
PROPOSITION l . An injeetion f:B—A induces an isomorphism
/ : Super [B] - Super [A].
PROPOSITION 2 . For any signed set A, we have a natural isomorphism of
Super[.A] with the tensor produet
Divp[A+]g
Wedge[yt~] 0 Symm[A
0].
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