TRANSACTIONS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 181, July 1973
THE POLYNOMIALIDENTITIES OF THE GRASSMANNALGEBRA
BY
D. KRAKOWSKI AND A. REGEV
ABSTRACT.
identities
By using
of the
1. Introduction.
T-ideal
and up to this
tive
and nilpotent
the Grassmann
of a given
point
algebras.
algebra
\l,
e
e2,
21
lm
of a basis
P.I.
'
—
in P.I.
algebras
the question
which
seems
algebra.
F of characteristic
/2
2
< ••• < i
M
is to describe
seems
to be quite
is generated
of
(exterior)
by the sequence
for E.
Hence
of
D =
We define the
£ D to be ttz. The
lm
for the case
e .e . + e .e . = 0.
I is a basis
diffi-
to generalization.
The Grassmann
777
■• • e .
the
for the commuta-
is answered
open
with the identity
1 ¡t a = e .
of polynomial
are the identities
of the Grassmann
1
¿"-ideal
The problem
cases
paper
• • ■ \ together
element
the
is computed.
algebra.
known
a method
Í1 ! U \e . • • ■ e . 11 < z', < i
length
algebra
In this
using
E over any field
elements
of codimensions
question
the only
2. The codimensions
algebra
theory
(exterior)
An important
of identities
cult
the
Grassmann
following
facts
ate immediate:
(1) If a £ D has
even
length
then
a £ Z{E),
the center
of E.
(2) If a, b £ D both have odd length, then ab = -ba.
Using
(1) and (2), one proves
(3) [[x, y], z] = 0 for any
x, y, z £ E,
where
[[,],]
is the Jacobi
polynomial,
[x, y] = xy - yx.
For any choice
those
indices
T2 elements.
of T2 elements
for which
Then
in D, [a.,
•••, a
a . is of odd length.
by (1) and (2) we have
Let
that
\, let
/ C j 1, • • • , « i denote
a £ S , the symmetric
aa
••• a a
group
on
= (/,(ff))t2 j • • • a ,
where /;fo) e { +1 \.
The
function
an ordered
o £ S
/ is therefore
set are in one-to-one
determines
a reordering
to fo , • • • , a ).
The reordering
by the one-to-one
Received
Now consider
o on jl,
• • • , n\
computed
as follows:
correspondence
since
the permutations
with the reorderings
of the T2-tuple (1, 2, • ■• , n) from its natural
an arbitrary
induces
correspondence,
by the editors September
AUS (MOS) subject classifications
yields
subset
I C\l,
a reordering
a permutation
■ • • , n\ and
of the subset
on /.
(Note
of
of the set,
order
a £ S .
/ and hence,
that
this
does
21, 1972.
(1970). Primary 16A38.
Copyright © 1973, American Mathematical Society
429
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430
D. KRAKOWSKI AND A. REGEV
not imply
that
mutation
ail)
C /.)
It can easily
is f ¡io~). For example,
il,
3, 5Í, then
the subset
(l,
3, 5) to (5, 3, l).
/ has
This
be shown
that
the sign
of this
if n = {l, 2, • • •, 5}, O = (been
reordered
corresponds
. .
induced
,
:) and
by o from the ordered
to the permutation
L
.
f
per/ =
triple
on I and hence
f.ia) = -1.
From
the above
2" x 72! matrix
of il,
S
we have
• • • , ?2S and using
and using
case
these
we shall
in ixj
2" x S 72 —> i+li.
—
write
shall
these
H as
indices.
F.
The
and ordering
the
indeterminates.
F[x]
d 72 of the 77-ideal
shall
of
the 777th
a e S 772.
\ o e S \ where
denote
Q
C F[x]
"- "~
the
tz! permutations
with
Ko)); ' I —
C i 1, • • • , m \ and
• • • xa
with
the 2" subsets
specifically
V (x) = Sp^ ix^
codimension
be concerned
by ordering
When dealing
H(m ' =--iff
1I
define
set of noncommutative
over
We shall
be determined
as row indices,
as column
As in (1) we shall
an infinite
'/:
H = if Âo)) which
x . £ ¡x ¡,
the free ring
is
V ix)
dime-
p n v 72(x)
"^
Lemma
('2.1).
polynomial
Proof.
Let
if and only
¿-„ ^ c
a
0 e ¿ n
Let
\d
identities
o
gix
o \
•••x„
Then
for the T-ideal
d
= rankp
By multilinearity,
on all the basis
and consider
o ji
sequence
algebra.
, • • • , x ) e V ix).
if g vanishes
x„
\ be the codimension
of the Grassmann
elements
the
a
er
gix
in D.
Write
as unknowns.
Let
Q of the
H n .
, • • • , x ) is in Q
gix.,
•••, x ) =
'<fl.!n_,
CD
7 2 —L —
and
substitute
giav
■■■,an)=
Z
o-eS
72
%aa---ao'
=
"
Z
o-eS
^(/f"^))«!
•■■ an
72
Z ^/I^Vi-v
creS
So g(*i>
/ C il,
• with the
/CÍ1,
• • • , 77¡. The
is tz! - rank,-.r H
72!,hence
I
■ • • , x ) is an identity
• • • , 721. This
ajares
n
d
yields
if and only if SCT£ s
a set
of 2"
2" x nl matrix
dimension
and this
linear
of the solution
some
notations
iiv---,i)=(1
For
o = ii ,,■••,
72+1'
(z ,,•••,
1
space
the dimension
to be used
in 72! unknowns
of the above
of Q
C\ V 72(x).
"^
zJ. — ,,
72,
z'7.,,,...,
l'
'
+1 '
Z
for o e S
72—1
z 72) e S 72', then
linear
Now
equations
dim V 72(x) =
H{n).
Q.E.D.
section:
indicate
, we define
o € S n,
H(n)) = tank,-
2 ■■■". \eS
i ) e S , io, n + l) shall
similarly
in this
= 0 for every
= H(n) = iffKo));
= n\ - dim,, (Q n V ix)) = n !- (77! - rankc
We define
n + l) e S .,;
equations
of coefficients
is also
aaf,nio~)
.
the permutation
io, n, n + l) e S
we indicate
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(¿,,
...
72+1
the replacement
r
• • •, i ,
If o =
of 72 by7
POLYNOMIAL IDENTITIES OF THE GRASSMANNALGEBRA
72 + 1 as
oin , n + l) = (z',,1 • • ■ , i.7 — i , n + I, i 1..,,
+ I
a permutation).
positions
of 77 and
Instead
2n~
Thus,
of working
and
H,(77)
with
M
of S
is not, " as it stands,
fofo", n + l), n) £ S +, exchanges
the
72 + 1 in {a, n + l).
submatrices
mutations
if a £ S , then
• ■• , i 77) (this
431
' of H
2"~
we define,
by induction,
' by choosing
and ordering
subsets
of jl, - - -, wfo This
a sequence
a subset
of 2"
x
of 2"~
per-
is done as follows:
Begin
with
(1)
,(L
and assume
that
AT"'
,72-1
1, 2,
has
defined
with
columns
indexed
, and rows indexed by / . C {1, • • • ,n\,
2" permutations
indexing
!
2" row indices
0
[io(k~2
,77-L
by1 aÍ7Í 6 S rt ,
f = 1, 2, •••, 2 72-1
The
of zVl(77 + 1) are
the columns
fo(k)
yik)
For the
been
for 1 < k< 2n~X,
\n~,n
+ l),n)
fot 2n~X < k < 2'
we have
72-1
h
(it follows
easily
for example,
!Ik
/
for 1 < k < 2
*< 2"
, U JT2+ 11 for 2"-1 < *
by induction
we obtain
M
that
from
/
M
is a row index
if and only if
1 £ J A.
Thus,
:
(fo2) _fo,fo
i(2)
1
-1
11,2}
(1,2, 3)
(2, 1,3)
1
1
.(3)
(1,3,2)
1
{1,21
-1
1,2, 31
(a) f\n\o)
For
For I £ I C\l,
-1
■• • , n\ and o £ S ,
=/<" +1)fo,T2 + 1) =/^1,^.n
+ 0.
1 £ I C jl, ••• , T2- l! and o £ S 72-1
(b) f\"-l)io) = f\" +X\a, n + 1, 72)= /J£Vijk. « + 1, «),
(0 f\-'\o)
1
-1
11,31
Lemma (2.2).
(3, 1, 2)
= f\"u¡X¡io, n + 1, n) = -f(?u\ln] n + ]lfo, n + 1, 22).
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'
432
D. KRAKOWSKIAND A. REGEV
For 1 e I C\l,
• ■■,n - l\ and o e S
(d) ffKo) = ff +XKoin~,n+ l),n),
^U\nu¡n]M=f\^oin%n
and
Proof.
The method
/Cil,
• • ■ , n].
set
a reordering
tion whose
to prove
sign
that
removing
of computing
Remove
the complementary
mining
il,
ff'io)
Lemma (2.3).
Proof.
The
zV!(r', 1 < r < n.
case
statements
a = ii.,
• • • , z ) those
the remaining
integers
reordering
determines
immediate.
o = ii.,
result
in
as detera permuta-
For example,
• ■• , i ) and note
the same
hence
■• • , i ) € S
letters
now become
i leaves
i , n + l\;
n = 1 (and
zVL"
2n~2
This
(a)—(e)
•••,!
to show
for
_
(z.,
Let
that
as removing
the two signs
are equal.
Sim-
of the lemma.
The rank of H(n) >2n~l.
n.
•• • , /
[i.,
the other
is true
Let /,,-•.,/
1
order.
Claims
\i ,,•••»
It is sufficient
We describe
set
Ko, n + l) we write
for all
ini,
natural
f,io).
= ff
yield
is as follows:
from the ordered
)l, • • •, ni - / from
arguments
/,io)
■• ■, n\ - I and consider
of their
is then
il, ■• ■, n, n + l\ - I from
ilar
+ l,n).
that
the rows
n = 2) is true
' by beginning
-, be the row indices
with
of M
by inspection.
M
Assume
and using
for M
U in i are the row indices
ate linearly
and hence
the result
two inductive
. Then I 1,,■••,
for /VL"\
independent
steps:
I 2n~2 -,,/.Ui
the row indices
M<" + 1> are""
V-..,
I^n_2, /, U {«!,...,
/ 2n—2 o U in + li,
Similarly,
let
a
/,ui«.
1
, •••, o
?2n_2 Uin!,
n+li,
•••,/
be the column
J, Uin+
2 n~2
indices
Ii,---,
, u in, « + 1(.
for M
. Then
those
for M(n) ate
io^X\n),...,io{2n~2\n),
io(XKnand for M
'we
r, «), «-
1), .-.,
(a(2""2)(n-
1", n), n - D
obtain
(cr(1), n, n+
1), •••,
(^(2"""2),
n, n + 1),
n- 2,
(cr(1)(n - 1", n), nio(l),
1, « + 1), ••-,
(?(2
Kn-
n+
(f7U
\n+
1, n), •••,
1", n), n-
,-.n—2.
(cr(1)(nPartitioning
1, n+ 1),
1, n),
^
L, n + 1), nn - 1, n), •••,
■■■, io{2"
io
' Kn
'in - 1~,
If n + 1), n - 1, n).
the index
sets
in this
manner
determines
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the following
division
for
POLYNOMIAL IDENTITIES OF THE GRASSMANN ALGEBRA
of Mi(n + l)
into
four quarters,
and of two of the quarters
into
+
8 subquarters:
_,'
+
ON
.—I
(N
« n
ii77+1
[1,21
1, 2, ...,
n-l\
fo, 72Í
U, 2, TZÎ
c
|1, 2, • • • , 72- 1, T2Í
,(77 + 1)
jl, 72+ 11
jl, 2, 72+ U
¡1, 2, • • • , 72- 1, 72 + 1|
jl,
72, 72 + 1|
m 77+1
j 1, 2, 22, n +11
1,2,
• • • , 72, 72+ lj
IV
The following
Sublemma.
facts
are a restatement
of Lemma
(2.2):
(i) I"+1 = IH" + 1 = M{n).
(ii) A =D =M{"-X).
(iii) C =-F = M("_1).
(iv) B = II".
(v) E = IV".
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433
434
D. KRAKOWSKI AND A. REGEV
Proof.
We prove
by the manner
row indices
only (iv);
of construction
/,•••,/
the rest
of the
_2 Ç [l,
n - l), ■ ■ , io
are similar.
/VF"''s,
•••»«
is the matrix
— l} and column
'in — 1, 72)72— l).
indices
II"
With the notations
Now
{a
determined
indices
B is determined
io
and by the column
fo
'in — 1, T2+ l), n - 1, 72). So we must show that for a £ S
/ Cjl,
...,72-
is exactly
Proof
|(72 + 1)
by the
n - I, n),
row
•••,
and
1|,
f\n){o{n- if«),»-
l) = f\n +X\oin - if« + 1), 72- 1, «).
the content
of the Lemma.
independent
and
'in - l,n),
by the same
indices
But this
'in - I, n + l),
above,
rows,
of Lemma
Since
I"
by induction.
from the lower
half
(2.2d)
= M
(using
half of M"
has
linearly
subtraction
of the upper
half
the Sublemma)
in the lower
half
of
IV" - II"
0
L=
a = {oin - 1, tt), 72 - 1).
, the upper
Upon rowwise
we obtain
with
0 [
■2zVi
77-I
Now
n"
I"
M(77)
.m"
By similar
arguments
upper
from the lower
half
I" = III" = M
half
'.
we obtain
/I"
M(72-1)
that
,(n)
AF"'
and
rows.
an identity
called
to be used
will
of /j-type
of the form x,
an algebra
The basic
subtraction
of the
II"
\
have
linearly
rows.
independent
Therefore
rows and hence
L,
and hence
AF"
it follows
, has
Q.E.D.
3. Codimensions
fying
by rowwise
IV" - II" >
IV" — II" = E — 3 has independent
independent
Therefore
the matrix
I
0
By induction,
IV",
algebras.
Let
• • ■ xrf = la
F be a field.
£S^ ^^
a^x^
An algebra
'"
x°d
satis-
wil1 be
of type / ..
reference
for this
section
be found
in the first
will be [l].
15 lines
The definitions
of § 1 of [lj;
that
is,
and notations
the definitions
of Vn fo), VU)ix),
UU){x).
As in Definition (1.5), [l], the codimension
n
n
T-ideal Q Ç F[x] is
V fo)
0 n V ix)
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d n of the
435
POLYNOMIAL IDENTITIES OF THE GRASSMANN ALGEBRA
Following
&
Lemma
(1.7)
*
7 in [l] with
0~
(x) = 0~- n V 72 ix) replacing
r
72
Wil, n) we have
°
Lemma (3.1).
V(t)+ UU)+Q
d =Z d
72
We shall denote
We shall
make
use
UU)+Q
dimPr iVlk)
+ (7(*)
+Q
)/(<7U)
+ p~ 72) as a.u>.
72
72
^72
72
(72)
'"
the space
(=1
of the short
"„-I^-SP^a
denote
'—•
notation
*<*„-] ' S<T1'""'
spanned
<p be the isomorphism
by all the
induced
yt "* xt + i> ' ' " ' y„-
(x ) = (x .,•••,
ff7z-lS = !l' ■••»'-
in - l)!
multilinear
Then
0(Vn_
j(y))
1—
<t —
< n.
in (x ).
y . —> x , ...,y
= Vn_ fx
■• • ,
l> ' + 1, ■■■, »il'
monomials
by the correspondence
i —* x„-
Let
x , • • -, x ) = (x,,
—►
x _,,
) and denote
<t>iQ
,(y)) byJ O
.Gc\).
It is obvious that x t Vn— ,(xj
= \Z(i)(x),
1
n— i J
~~ n— I
t
1
t
n
Let
the
image
and that
*tQn-fit)ÇQnix).
Theorem
(3.1).
codimensions.
For the proof
Let
£ V 72 Kx).
• • • ,p
we shall
will
so that
ik = uf,
Denote
need
the
iM(x)= (x
of type
three
lemmas
order
x ,
>1
1 z-j
)(x
\ its sequence
and the following
of
preliminaries:
and let the monomial
which
of appearance
••• x ,
\d
.
s = n - k + 1 indices
by x. 1 i according
al
] ,, with
< id — l)"~
1 < k < n,
to their
be indicated
1
n, d
integer,
according
indices
A be an algebra
for any
n be any positive
&n
p.,
Let
Then,
satisfy
<W(x) = x ,xa
in the monomial
to their
x,
order
• • ■ x.,
M2 2[
2,2
2
a,k —
> k byz
'
Mix).
of appearance
) • • • (x
x
zzs j]
The
after
■• • x
other
p.'
srj
).
The mapping
induces
Using
an isomorphism
this
notation
yi ^(v,xi1
■•• xir )
y
—. (X,, X
■'s
fi si s,
•••
i/7: V (y)
we have
—>V ix)
the following
X
which
lemma,
s
7
is an into
whose
linear
proof
transformation.
is immediate:
Lemma (3.2). (a) Mix) e </z(l^ ' \y)) Ç V^*>(x).
(b) ifAU(sX)iy))ÇUlkKx).
(c) </>(0
(y))CO
(x).
' —s '
— —n
Lemma (3.3).
Let A be an algebra of type J ,. Let d <n and 1 <k <n - d + 1.
Then a.k\ = 0.
(72)
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436
D. KRAKOWSKI AND A. REGEV
Proof.
monomials
above.
We show
spanning
V<nk) C ifo *° + Q . It is sufficient
fo77 '.
°
Let
Al(x) = x,x„
fe
zé<T2-zi+l=»zi<T2-zé
y j ' ' ' y ¿ = ^o- ,^1
+ l=s.
aCr)'(T
yd+\
'"
Each
term and hence
yA ODtaining
••• x„crn £ V n
Since
' ' ' y o- • Since
an identity
ex 2
to prove the result
ix),
A is of type
and
for A of the form yt •••ys
side
U(sX)iy), so that yx ■■■ys £ U^Ky) + Qsiy).
both
= ^a
Proof.
By definition,
the isomorphism
r
Multiplication
F(
on the left by x, -induces
,
*i.F
k
,.
a
, = dim,;
onto x.Q
,(x.)k + x,0
k^-
77— 1
-
j(y) + Q
of V
, (x.)
onto
Hence
,(x.)«
77— 1
vU) + ua)(x) + g fo)
> dimF -2-2-12—
-
-
Proof
Q.E.D.
p(y))/Qn_ p(y). Applying
an isomorphism
_ jfo^).
v(fe)(x)+ e fo)
ß„W
in
,(x.)Ze +Q*-72—1,(x,))/0
,fo,).
72—1
k
~n—1
zé
> dimF -2-3_
F
' ' ' Va
contained
72-r
^n_ ¡ = dimF (V
Q _ ,(x,)
by
Hence
cp we obtain d 72—1, =dim,,fo
r"
'(x) sending
sides
¿x a0yal
is obviously
Mix)= tfAiyi... yj e ¿iU^Xy)) + iftiQsiy))C fo*°fo)+ Qix).
Lemma (3.4). For I < k <n, a^l <
vj
be as
~
/ , it satisfies
zi < s we can multiply
the sum of the right-hand
for the
F
= a\k\.
U(k\x) + ß (x)
Q.E.D.
M
of the Theorem.
< (d — l)n~
If n —
< d - 1 we are finished, ' for in this case d « —
< n!
n > d and induct: ¿/
, < (¿/ — l)n"
. By the lemmas above
. Assume
—
—
¿77 =£«,*}
=
*-;
(77)
Te= l
Corollary.
Lei
n —i —
Z
^-
k = n—d + 2
«<*><U-1)¿
(77) -
{d \ be the codimension
y
.1 <UI)"-1.
-
72-
sequence
Q.E.D.
^
for the Grassmann
algebra
E. Then d 72 = 2""1.
Proof.
d
< 2n~
Since
E satisfies
. Using
Lemmas
[[x, y], z],
(2.1)
E is a / , type
and (2.3)
we conclude
algebra
that
d
and hence
> 2n~
, and hence
d 77 = 2"_1.
4. Applications.
implies
linear
ties
It is well
the existence
part
of a homogeneous,
of a T-ideal
in the ideal.
known
we shall
that
the existence
multilinear
mean the vector
For a polynomial
/ £ F[x]
of a polynomial
identity
space
we shall
[2].
By the multi-
of all multilinear
denote
identity
identi-
the T-ideal
gener-
ated by / as Tif).
The following
literature
lemma
and so we include
is known
but its proof
is difficult
it here.
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to extract
from the
437
POLYNOMIAL IDENTITIES OF THE GRASSMANN ALGEBRA
Lemma
whose
(4.1).
Let
multilinear
F have
parts
characteristic
are the same.
0, and
Then
Proof. Let MÍP) be the multilinear
¿(x j, • • • , x ) £ F[x]
have
and show
' = t, and consider
define
f e Q.
Let
the sequence
degree
The ¿-tuples
thus
We shall
Write
show
/ = 2
/
part of
of x . = d.,
P. So MiP) = MÍQ). Let
and
th = max \d .{. We assume
A = \h e P\ t, <t\.
• • • , r ) where
Assume
that
that
defined
are completely
/ £ Q by applying
r. = the number
field,
each
/ itself
x . has
/ e P
For any h e A
of variables
of
one defines
giu,
Hence
polynomials
We shall
t in /.
that
■••,
But giu,
°
x^
has
degree
x 72) - 'fiu,
x,,I
•••,
u, x.,•••,
Z
t.
tuple induction,
Let
comparable
variable,
and
e Q and hence
hence
in
In the multilinearization
- l)fiu,
the T-ideal
...
and precedes
g e Q and so / £ P.
P denote
f
multilinear,
x 72) -/(zz,
x 72 ) = 2(2'~
• ■• , r ) then g —»rig) = (sj,
g is lexicographically
that
If t = I f is already
char F / 0 and £ > 1, 2(2'" 1 - l) / 0 and so / £ T(g).
If / —* r(/) = (fj,
in every
show
■■ ■, x)
+ v, x-,¿
g° £ T(/).
£ P.
lexicographic
[2, p. 224]
v, x2,
= fiu
J
/
by the right
induction.
is homogeneous.
degree
So t > 1 and we may assume
process
ordered
¿-tuple
as a sum of homogeneous
F is an infinite
we may assume
p.
rih) = ir.,
be two T-ideals
; in h.
order.
since
degree
P, Q C F[x]
P = Q.
x„,¿ •■•,
x 72).
x ¿ , • ■• , x 72 ) and since
Hence T(/) = 7(g).
, s ) with s( < r(.
/ in the order,
Now g £ P,
hence
by the k-
Q.E.D.
of identities
of the Grassmann
algebra
E,
and
Q =
r([[x, y], *]).
Theorem
/2
(4.1).
the codimensions
Proof.
Since
For the Grassmann
algebra
of P = the codimensions
E satisfies
the identity
E over a field
of Ti[[x,
and these
are the codimensions.
Hence,
y], z]) = P,
[[x, y], z] we have
P 72(x) = P n V72ix) —A Q
nVW.OW-.
*72
*-72
F of characteristic
that
> dimP(K
C
—
/p
7272—
by Lemmas
) > dimF(V
C
dim,-.
P 72 = dim^.
0 = n\ - 2"~
r
r —n
Now P Tjp
Corollary.
=> P =0
.
P
0 C P.
= Q .
Hence
dimJV
/P ) > dimP(V
/P 72),
r
72 ^ 72 —
r
72
(2.1),
(2.3)
and the corollary
to Theorem (3.1),
2"-1
and
IP 77) >
2"-1,
—
72
.
Q.E.D.
// char F = 0 then P = Q.
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438
D. KRAKOWSKI AND A. REGEV
Proof.
variables
Q
in
and
P and
P
are just
Q.
Since
the multilinear
these
homogeneous
are T-ideals
polynomials
in tz
M{P) = M{Q) and so by Lemma
(4.1), P = Q. Q.E.D.
Let
alized
and
T = ([[•■• [fop x2], xA •■■], x ]) be the T-ideal
Jacobi
je
polynomial.
Let
S be its sequence
(3.1), "„=fo-
F[x]/T
be the universal
of codimensions.
Then
generated
algebra
by the gener-
for this
we conjecture
that,
identity
as in Theorem
1)1W- l)"-d+1.
REFERENCES
1. A. Regev,
2. N. Jacobson,
Existence
of identities
Structure
of rings,
in A ® B, Israel J. Math. 11 (1972), 131 —152.
2nd rev.
37, Amer. Math. Soc, Providence, R. L, 1956.
ed.,
Amer.
Math.
Soc.
Colloq.
PubL,
MR 18, 373.
THE WEIZMANN INSTITUTE OF SCIENCE. REHOVOT, ISRAEL
Current
Detroit,
address
Michigan
Current
Los Angeles,
address
(D. Krakowski):
Department
of Mathematics,
Wayne
State
University,
48202
(A.
California
Regev):
Department
of Mathematics,
90024
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University
of
California,
vol.