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