2
THE SUPERALGEBR A Super[A ]
Much of the work i n this chapter goe s in showing that th e structur e o f a Hopf
algebra ca n b e give n t o th e superalgebr a Supe r [A] b y settin g
A(a)W= ^2
aU) ®a(k)i
aeA
+,
j+k=i
Aa = l 0 o + o 0 l , o G i ° U i " ,
and extendin g A s o tha t i t i s a n algebr a homomorphis m fro m Supe r [A]
to Super [A © A]. Th e onl y delicat e poin t arise s i n th e commutatio n rul e
(1 0 a) (b 0 1) = b 0 a whe n bot h a an d b ar e negativel y signed . Becaus e
of this feature , th e superalgebr a Super[ A 0 A] cannot b e identifie d wit h th e or -
dinary tenso r produe t Super[A ] 0 Super[A ] o f commutative algebra , bu t rathe r
it i s a "signed " tenso r product .
In geometri c an d combinatoria l interpretations , th e neutra l element s will no t
directly appear ; the y appea r indirectl y a s "scalars. " Remarkably , positivel y
signed variables give the symbolic representation o f skew-symmetric tensors , an d
negatively signed variables give the symbolic representation of Symmetrie tensors.
Combinatorially, positivel y signe d variable s relat e t o th e algebr a o f multiset s
(or bosons) , an d negativel y signe d variable s relat e t o th e algebr a o f set s (o r
fermions).
The material i n this chapter offer s n o great novelty . Reader s acquainte d wit h
the techniques of Hopf algebras may simply skim over the definitions an d proeeed
to th e nex t chapter .
1. Definitions . Le t A b e a set . W e denot e b y Mon(A ) th e free monoid
generated by A. W e recal l tha t th e element s o f Mon(A) ar e finite sequence s o f
elements i n A. I f w £ Mon(A) wit h w(l) = xi , w(2) = X2,...,w(n ) = x n,
then w e shall cal l w a word and denot e i t b y w = X\X2 xn. Tw o suc h word s
u) = x\X2 xn an d w' = yiy2 ym ar e equal when n m an d x\ = t/i , xi
y2,--,xn =|/n -
If w = x%X2 - xn i s a wor d i n Mon(A) , w e shal l cal l X\X2 * xn it s display
and n it s length. Th e word s having length 1 will be identifie d wit h th e element s
of A. W e shall allo w w t o b e th e empt y sequence , i n this cas e takin g n t o b e 0 ,
and denotin g thi s word b y 1. (T o avoid confusion , w e shall alway s assum e tha t
I*A.)
The product o f tw o word s w X\X2 xn an d w
f
= y\y2 * 2/m is th e wor d
wwf
= X\X2 xnyiy2 t/m- Endowe d wit h thi s Operation , Mon(A) i s an asso -
ciatiave (bu t no t commutative ) semigrou p wit h identity .
A multiset o n a se t A i s a funetio n m: A Z + wher e Z + i s th e se t o f
nonnegative integers . Fo r a A , th e intege r m(a) i s calle d th e multiplicity o f
the elemen t o i n th e multise t m . Ther e i s a usefu l wa y t o visualiz e multisets ,
which w e shal l ofte n tacitl y appea l to . Suppose , fo r example , tha t m(a) =
2, m(b) = 1, m(c) = 3 and tha t m(x) = 0 for al l other x A. The n w e describ e
the multise t b y th e displa y {a , 6, c, a, c, c}.
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