HIGHER CONJUGATION COHOMOLOGY
IN COMMUTATIVE HOPF ALGEBRAS
M. D. Crossley and Sarah Whitehouse
22nd July 1999
Let A be a graded, commutative Hopf algebra. We study an action of the sy*
*mmetric
group n on the tensor product of n - 1 copies of A; this action was intro*
*duced by
the second author in [8] and is relevant to the study of commutativity co*
*nditions
on ring spectra in stable homotopy theory [6]. We show that for a certain*
* class of
Hopf algebras the cohomology ring H*(n; An-1 ) is independent of the cop*
*roduct
provided n and (n - 2)! are invertible in the ground ring. Then, by choo*
*sing a
sufficiently simple coproduct, we are able to deduce significant informat*
*ion about
the n invariants of An-1 , including dimensions and algebra structure.
1. Introduction and Main Results
Let A be a graded, connected, unital, counital, associative, coassociative H*
*opf
algebra. In section 8 of [5] it was shown how A has a `conjugation' or `antipod*
*e' O
satisfying the equality
O (1 O) O = j O ffl;
where and are the product and coproduct and j and ffl are the unit and counit.
In particular, O(1) = 1 and, for x of positive degree,
X
O(x) = -x + x0iO(x00i)
i
P
where (x) = x 1 + 1 x + ix0i x00i. If A is commutative then O2 = 1 and so
gives an action of 2 on A.
The second author extended this in [8] by providing, for each n > 2, an acti*
*on
of the symmetric group n on An-1 when A is commutative. If oei denotes the
transposition i $ i + 1, then n is generated by oe1; : :;:oen-1 , and the actio*
*n on
An-1 is given by:
oe1= [( 1) O (O )] 1n-3
oei= 1i-2 [(1 ( O ( 1)) 1) O ( O )] 1n-i-2 if1 < i < n - 1
oen-1= 1n-3 [(1 ) O ( O)]
Note that each oei acts multiplicatively, as does O in the case n = 2.
______________
1991 Mathematics Subject Classification. 16W30, 57T05, 20C30, 20J06, 55S25.
Typeset by AM S-T*
*EX
1
2 M. D. CROSSLEY AND SARAH WHITEHOUSE
Thus if A is commutative, as we will henceforth assume, we have a multiplica*
*tive
action of n on An-1 for each n 2. While it may seem unusual to have n acting
on an n - 1-fold product, this action does arise quite naturally in stable homo*
*topy
theory as will be explained in Section 3. Motivated by this application, we wis*
*hed
to calculate the cohomology of n with coefficients in An-1 . However, in [3] we
saw how complicated this calculation could be when we attempted it for n = 2 and
for A an object familiar to algebraic topologists: the mod 2 dual Steenrod alge*
*bra.
Since these n actions explicitly involve O and , and the former involves the
latter, it would be desirable if we could make the coproduct as simple as poss*
*ible.
The following theorem, which is our main result and is proved in Section 2, giv*
*es
conditions under which we can do this without changing the cohomology ring that
we wish to calculate.
Theorem 1. Let A be a graded, connected, coassociative Hopf algebra over the r*
*ing
R. Suppose that A is isomorphic, as an algebra, to a tensor product of associat*
*ive
Hopf algebras, each of which has just a single algebra generator. Let "Abe this*
* tensor
product, considered as a Hopf algebra. If n and (n - 2)! are invertible in R t*
*hen
there is an isomorphism of algebras
A"n-1 - ! An-1
which commutes with the n action, thus inducing an isomorphism of cohomology
rings
H*(n; "An-1 ) H*(n; An-1 ):
Note that by assuming that the underlying algebra of A is a tensor product of
associative monogenic algebras, we automatically have that A is commutative (and
associative).
The importance of this theorem rests on the fact that the coproduct in A" wi*
*ll
generally be much simpler than that in A since all the generators of "Aare prim*
*itive.
Thus calculations in A"will be simpler than in A and in the rest of this sectio*
*n we
consider results that would be hard, if not impossible, to obtain by looking at*
* A
directly.
In particular, if A is a graded, connected, commutative, biassociative Hopf *
*al-
gebra, of finite type and R is a perfect field then, by the Borel-Hopf theorem *
*([5]
Theorem 7.11), A satisfies the hypotheses of Theorem 1. Our Theorem is more
general than this in that we can work over a ring instead of a field, and we do*
* not
need A to be of finite type so, for example, A = Z=8[x1; x2; : :]:, where |xi| *
*= 2 for
all i, satisfies the hypotheses of Theorem 1, but not of the Borel-Hopf theorem.
The hypothesis that n:(n - 2)! be invertible is rather curious and we will d*
*iscuss
this further at the end of Section 2.
Of course, if we make the stronger assumption that n! is invertible in R, th*
*en
the cohomology ring is zero in positive degrees, but even in this case the theo*
*rem
is of significance since it greatly simplifies the calculation of H0(n; An-1 )*
* =
(An-1 )n = {x 2 An-1 | oex = x for alloe 2 n}. In particular, if R is a f*
*ield
and A is a polynomial algebra, then the theorem puts this calculation into the *
*realm
of classical invariant theory. This is because n acts multiplicatively on "An-1*
* and
preserves the R-subspace Q A"n-1 spanned by the generators. So we have n
HIGHER CONJUGATION COHOMOLOGY 3
acting on a vector space Q, and the action on A"n-1 is just the action on the
polynomial algebra R[Q] (= "An-1 ) induced from this n action on Q.1
In fact one does not necessarily need A to be polynomial. For example, Moli*
*en's
theorem can be re-worked as follows.
Theorem 2.
1) Suppose R is a field whose characteristic is either 0 or coprime to n!. Le*
*t A
be generated by x1; x2; : : :of positive degrees d1; d2; : : :and heights h*
*1; h2; : : :
(chosen from N [ 1). Then the Poincare series for (An-1 )n is given by
X Y det(1 - gtdihi)
p(t) = _1_n! _______________di;
g2n i det(1 - gt )
where, in the i-th factor on the right, the determinants are those of the g*
*iven
operator acting on the space .
2) If R is a field whose characteristic divides n!, then this Poincare series *
*bounds
that for (An-1 )n coefficient-wise.
Proof. The proof of the first statement is a simple modification of the standa*
*rd one
as found in, e.g. [2], [7].
The proof of the second statement begins with the remark in [2] that in the
modular case, the series above gives the Poincare series for the multiplicity *
*of the
trivial representation as a composition factor in A"n-1 . Thus the Poincare s*
*eries
given above bounds that for (A"n-1 )n . To complete the proof we will show t*
*hat
the n-invariants in An-1 can be mapped injectively into those for A"n-1 .
Given an element of An-1 we write l() for the `least decomposable part' *
*of
. To be precise, filter An-1 by powers of the augmentation ideal, so that F0*
* is
the positive degree part of An-1 and Fk = F0 . Fk-1 for all k > 0. We may th*
*en
write as 0 + 1 + . .,.where i is the sum of those monomials lying in Fi but
not in Fi+1, and l() will be k where k is the least integer such that k 6= 0. *
*The
key point is that if is n-invariant in An-1 then l() is n-invariant in A"n-*
*1 .
This follows from the fact that if x is an indecomposable in An-1 , then oei(*
*x) in
A"n-1 is precisely the least decomposable part of oei(x) in An-1 .
We claim that there is some choice of basis '1; '2; : :f:or (An-1 )n such*
* that
l('1); l('2); : : :are linearly independent in (A"n-1 )n , enabling us to de*
*fine an
injection (An-1 )n ,! (A"n-1 )n by 'i 7! l('i). To prove this, let OE1;*
* OE2; : : :
be any basis for (An-1 )n . We will inductively modify this basis until we *
*have
l(OE1); l(OE2); : : :linearly independent. Clearly l(OE1) on its own is linea*
*rly inde-
pendent, so now assume that l(OE1); : :;:l(OEm-1 ) are linearly independent. *
*If this
set remains linearly independent when we add l(OEm ) to it then there is noth-
ing to do and the inductive step is complete. If not, then there is a relation
a1l(OE1) + . .+.am l(OEm ) = 0, with am 6= 0. We replace OEm by a1OE1 + . .+*
*.am OEm ,
noting that the set OE1; OE2; : :;:OEm ; : :s:till forms a basis for (An-1 )n*
* . If l(OEm )
is now linearly independent of l(OE1); : :;:l(OEm-1 ) then the inductive step *
*is com-
plete. If not then, again, there is a relation b1l(OE1) + . .+.bm l(OEm ) = 0*
* and we
______________
1Ian Leary points out that the n representation Q is a tensor product of co*
*pies of the well-
known n - 1-dimensional n representation V constructed as follows. Take an n*
*-dimensional
vector space with basis e1; : :;:en and let n permute the basis. Then V is the*
* subspace spanned
by e1 - e2; e2 - e3; : :;:en-1 - en. There is one copy of V in Q for each gene*
*rator of A.
4 M. D. CROSSLEY AND SARAH WHITEHOUSE
replace OEm by the corresponding linear combination of OEi's. We repeat this *
*pro-
cess until we obtain some OEm such that l(OE1); : :;:l(OEm ) are linearly inde*
*pendent.
This must occur within a finite number of steps because at each stage l(OEm ) is
replaced by something of strictly higher filtration and in each degree the filt*
*ration
is finite. Moreover l(OEm ) cannot ever be zero for this would imply a relation*
* among
OE1; : :;:OEm .
By way of example, if n = 2 then the first part of Theorem 2 leads to the
following explicit formulae.
Corollary 3. If R is a field of characteristic different from 2 and A is gener*
*ated
by x1; x2; : :o:f positive degrees d1; d2; : :a:nd heights h1; h2; : :,:then th*
*e Poincare
series for A2 is
Q Q
(1 + tdi)(1 - tdihi) + i(1 - tdi)(1 + tdihi)
p(t) = __i__________________________________________2:Q 2d
i(1 - t *
*i)
If A is polynomial this simplifies to
Q Q
(1 + tdi) + i(1 - tdi)
p(t) = __i______________________2:Q 2d
i(1 - t i)
In fact, in the case n = 2 Theorem 1 leads to the following `model' for the
conjugation invariants.
Theorem 4. Suppose A is a graded, connected, coassociative Hopf algebra which,
as an algebra, is a tensor product of associative monogenic Hopf algebras and s*
*up-
pose that 2 is invertible in the ground ring. Let A2 denote the subalgebra of *
*conju-
gation invariants, and let AE denote the subalgebra of A spanned by the monomia*
*ls
whose exponents sum to an even number. Then there is an isomorphism of algebras
A2 AE :
This is a simple consequence of Theorem 1 and the fact that O(x) = -x if x is
primitive.
We note that if p > 2 then Theorem 4 along with Corollary 3 completely solves
the `conjugation invariants' problem for the mod p dual Steenrod algebra, in ma*
*rked
contrast to the partial solution [3] available when p = 2.
2. Proof of Theorem 1
We wish to construct an isomorphism of algebras f : A"n-1 ! An-1 that
commutes with the n-action. Thus there are three components to the proof :
we must define a multiplicative map, show that it is a bijection, and show that*
* it
commutes with the group action. We will begin by briefly discussing the signifi*
*cant
features of each of the three components. It will be useful, however, to first*
* fix a
set of generators for An-1 and A"n-1 . These objects are identical as algebr*
*as so
the same set of generators will suffice for both. Let J be a set of generators *
*for A.
For each x 2 J and for 1 i n - 1, define x[i] 2 An-1 by
x[i] = 1i-1 x 1n-1-i :
HIGHER CONJUGATION COHOMOLOGY 5
Then a set of generators for both An-1 and A"n-1 is given by
{x[i] | x 2 J and 1 i n - 1}:
Now, to define a multiplicative map f : A"n-1 ! An-1 we will construct a
map on these generators and extend it multiplicatively. This will give a well-d*
*efined,
multiplicative map provided that we show that f preserves all relations which h*
*old
among the generators of "An-1 . Since "Ais a tensor product of monogenic algebr*
*as
we know that the only relations in A"are of the form xh = 0 for some generator x
and positive integer h and it follows that the only relations in "An-1 are of *
*the form
x[i]h = 0. Moreover, h cannot be chosen arbitrarily, because R[x]=(xh) is assum*
*ed
to be a Hopf algebra (see the discussion in [4] x1-3). To be precise, h must be*
* such
that hj = 0 in R for 0 < j < h. With such an h we have that, if g1; g2; : :a:*
*re
multiplicative, e.g. elements of n, then yh = 0 implies (r1g1(y)+r2g2(y)+. .).h=
0 for any r1; r2; : :i:n R. In light of this, we will set f(x[i]) = aix[1], whe*
*re each
ai is an element of the group algebra Rn. Since the height of x[1] is the same *
*as
that of x[i], the above argument shows that f preserves all the relations.
To show that f is bijective, it suffices to show that the map induced on the
modules of indecomposables
______(A"n-1__)+______- ! ______(An-1__)+_______
(A"n-1 )+ . (A"n-1 )+ (An-1 )+ . (An-1 )+
is a bijection. Here the notation ( )+ denotes the augmentation ideal, i.e. *
*the
positive degree part of the algebra. We will show, at the end of the proof, tha*
*t this
induced map is a scalar multiple of the identity : for a generator x[i] we have
f(x[i]) x[i] mod (An-1 )+ . (An-1 )+
for some 2 R, and that the hypotheses ensure that is invertible so that the
induced map is a bijection.
Most of the work will go into showing that f commutes with the n action. To
explain further let us examine exactly what conditions must hold for f to commu*
*te
with the group action. To do this we need to understand how n acts on "An-1 but
because the generators x are primitive in A"it is easy to see what oekx[i] is, *
*working
directly from the definition. For example, if k = i + 1 then oekx[i] = x[i] + x*
*[i + 1].
From this we can see that for f to commute with the n action, the following
relations must hold for all generators x:
8
>>>f(x[i]) ifi < k - 1
>> -f(x[i]) ifi = k
>>>
>: f(x[i - 1]) + f(x[i]) ifi = k + 1
f(x[i]) ifi > k + 1
We will ensure these relations hold by making appropriate choices of the elemen*
*ts
of the group algebra used in the definition of f. For example, if i > 1, we wi*
*ll
ensure that oeif(x[i]) = -f(x[i]) by defining ai = (oei - 1)ai-1, i.e. f(x[i])*
* =
(oei-1)f(x[i-1]). Since oe2i= 1, we have the desired result that oeif(x[i]) = -*
*f(x[i]).
6 M. D. CROSSLEY AND SARAH WHITEHOUSE
Having thus outlined the strategy, we now present the details.
For each generator x of A", let
f(x[1]) = (oe1 - 1)Tnx[1];
where oe1 is interpreted as O if n = 2, and Tn 2 Rn is defined as follows. Let
Gn be the subgroup of n generated by oe2; : :;:oen-1 . Then Tn is the sum of t*
*he
elements of Gn. If n = 2 then we understand Tn as 1.
Then oe1f(x[1]) = -f(x[1]), which, for n = 2, is all that is needed to show *
*that
f commutes with the 2 action. For n 3, we also have that oekf(x[1]) = f(x[1])
for all k > 2, since, for such k, oek commutes with oe1 and oekTn = Tn; this la*
*tter
fact is true because oek 2 Gn and so left multiplication by oek simply permutes*
* the
elements of Gn, leaving their sum, Tn, unchanged.
Now oeif(x[i - 1]) should equal f(x[i - 1]) + f(x[i]), so we must have
f(x[i]) = (oei- 1)f(x[i - 1]) fori > 1:
Thus, having defined f(x[1]) above, we now use this formula to define f(x[i]) i*
*nduc-
tively for all i and then extend multiplicatively to define f on the whole of A*
*"n-1 .
It follows that oeif(x[i]) = -f(x[i]) for i > 1. Furthermore, if 1 i < k - 1,*
* we
also have that oekf(x[i]) = f(x[i]) since oek commutes with oei when i < k - 1 *
*and
oekf(x[1]) = f(x[1]) if k > 2.
Next we consider oeif(x[i + 1]):
oeif(x[i + 1]) - f(x[i]) - f(x[i + 1])
= [(oei- 1)(oei+1 - 1) - 1](oei- 1)(oei-1 - 1) . .(.oe1 - 1)Tnx[1]
= (oeioei+1oei- oeioei+1 - oei+1oei+ oei+ oei+1 - 1)(oei-1 - 1) . .(.oe1 - *
*1)Tnx[1]
= (oei+1oei- oei+ 1)(oei+1 - 1)(oei-1 - 1) . .(.oe1 - 1)Tnx[1]
= (oei+1oei- oei+ 1)(oei-1 - 1) . .(.oe1 - 1)(oei+1 - 1)Tnx[1]
and this is zero because oei+1Tn = Tn by the same argument as above (oei+1 being
in Gn).
Finally, if k < i - 1, then
oekf(x[i])= oek(oei- 1) . .(.oek+2 - 1)f(x[k + 1])
= (oei- 1) . .(.oek+2 - 1)oekf(x[k + 1])
= (oei- 1) . .(.oek+2 - 1)(f(x[k]) + f(x[k + 1]))
= (oei- 1) . .(.oek+3 - 1)(f(x[k + 2]))
= (oei- 1) . .(.oek+4 - 1)(f(x[k + 3]))
. . .
= f(x[i]):
Thus we see that oekf(x[i]) is as required for all i; k, i.e. f commutes wi*
*th the
n action.
We now finish by showing that f is a bijection on the modules of indecompos-
ables. Note that since f(x[i]) = (oei-1)f(x[i-1]) for i > 1 and (oei-1)(x[i-1])*
* x[i]
HIGHER CONJUGATION COHOMOLOGY 7
modulo decomposables, it suffices to prove that f(x[1]) x[1] modulo decompos-
ables, for some unit 2 R. For the remainder of the proof, the symbol will
denote equivalence modulo decomposables. First we need to calculate Tn(x[1]):
n-1X
Tn(x[1]) (n - 2)! (n - i)x[i]:
i=1
This is proved by induction using the fact that
Tn = (1 + oen-1 + oen-2 oen-1 + . .+.oe2 . .o.en-2 oen-1 )Tn-1 ;
i.e. the summands of the left hand factor form a set of coset representatives f*
*or Gn
over Gn-1 . This is easily seen to be the case since, under the natural action *
*of n
on {1; 2; : :;:n}, the permutations oej . .o.en-1 each map n differently.
Now, 8
>< -2x[1] ifi = 1
(oe1 - 1)x[i] > x[1] ifi = 2
: 0 ifi > 2:
Hence
f(x[1]) (oe1 - 1)Tnx[1]
n-1X
(oe1 - 1)(n - 2)! (n - i)x[i]
i=1
(n - 2)!(-2(n - 1)x[1] + (n - 2)x[1])
-n:(n - 2)!x[1]:
So, as long as n:(n - 2)! is invertible in the ground ring R, f gives a bijecti*
*on on
the indecomposables. This explains the provenance of the curious condition on R
in Theorem 1 and completes the proof of that theorem.
It remains unclear to what extent the hypothesis that n:(n - 2)! be invertib*
*le is
necessary. If n = 2 and R = Z=2 then the conclusion of the theorem is definite*
*ly
false (see [3] for an example). Furthermore, calculations with Magma have revea*
*led
a number of other examples where, if n is not invertible, the invariants, or the
higher cohomology, differ when one changes the coproduct. However, we have not
found any cases where n is invertible in R and where the conclusion of the theo*
*rem
does not hold.
3. Why should n act on An-1 ?
Let E be a ring spectrum and let A = ss*(E ^ E), the set of `co-operations' *
*in
the cohomology theory associated to E. (For example, if E is the Fp Eilenberg-M*
*ac
Lane spectrum then A is the mod p dual Steenrod algebra.) If E is sufficiently *
*nice
then A is a commutative Hopf algebra (more generally it is a Hopf algebroid). T*
*he
conjugation map, O, on A is precisely the map induced on ss*(E ^ E) by switching
the factors in the smash product E ^ E ([1] Lecture 3). Analogously, we can take
the homotopy of a smash product of n copies of E, ss*(E ^ . .^.E), and there is*
* a
natural action of n induced by permuting the factors. But ss*(E^n ) = E*(E^n-1 *
*),
the cohomology of an n - 1-fold product of copies of E and, for suitable E, thi*
*s is
isomorphic to An-1 . Thus there is an action of n on An-1 and it is shown in
[8] that this action satisfies the formulae given at the start of this paper.
8 M. D. CROSSLEY AND SARAH WHITEHOUSE
Acknowledgements
The first author was supported by a grant from the Max-Planck-Institut f"ur
Mathematik and would like to thank this organization for its support and hospit*
*ality
and for paying for a visit by the second author. We would both like to thank Ran
Levi for his comments on Molien's theorem.
References
1. J. F. Adams, Lectures on generalised cohomology, Springer Lecture Notes in
Mathematics, 99, Springer-Verlag, Berlin-Heidelberg-New York, 1969, 1-138.
2. D. J. Benson, Polynomial invariants of finite groups, London Math. Soc.
Lect. Note Series 190 (1993).
3. M. D. Crossley and Sarah Whitehouse, On conjugation invariants in the dual
Steenrod algebra, to appear in Proc. Amer. Math. Soc.
4. R. Kane, The Homology of Hopf spaces, North Holland Math. Library 40.
(1988).
5. J. Milnor and J. Moore, On the structure of Hopf algebras, Ann. Math., 81
(1965), 211-264.
6. Alan Robinson and Sarah Whitehouse, -homology of commutative rings and
of E1 -ring spectra, Warwick preprint 76/1995.
7. L. Smith, Polynomial invariants of finite groups, A. K. Peters Ltd. (1995*
*).
8. Sarah Whitehouse, Symmetric group actions on tensor products of Hopf al-
gebroids, in preparation.
Max-Planck-Institut f"ur Mathematik, P.O. Box 7280, D-53072 Bonn, Germany.
E-mail address: crossley@member.ams.org
Departement de Mathematiques, Universite d'Artois - P^ole de Lens, Rue Jean
Souvraz, S. P. 18 - 63207 Lens, France.
E-mail address: whitehouse@euler.univ-artois.fr