LOW DIMENSIONAL COCOMMUTATIVE CONNECTED HOPF ALGEBRAS Gregory D. Henderson Abstract. William M. Singer's theory of extensions of connected Hopf algeb* *ras is used to give a com- plete list of the cocommutative connected Hopf algebras over a field of po* *sitive characteristic p which have vector space dimension less than or equal to p3. The theory shows tha* *t there are exactly two noncommutative non-primitively generated Hopf algebras on the list, one of* * which is the Hopf algebra corresponding to the sub-Hopf algebra of the Steenrod algebra generated by* * P1 and Pp. The commu- tative Hopf algebras are found using Borel's theorem and the primitively g* *enerated Hopf algebras using restricted Lie algebras. Introduction. In this paper we study low dimensional cocommutative connected k-Hopf algebra* *s for k a field of positive characteristic p using William M. Singer's theory of extensions of con* *nected Hopf algebras [2]. Specifically, any finite dimensional cocommutative connected k-Hopf algebr* *a occurs in a central extension A ! C ! B where B is a cocommutative connected k-Hopf algebra of vect* *or space dimension strictly less than that of C and A is polynomial on one generator tru* *ncated at height p or exterior on one generator. Singer describes a cohomology group H3(B; A) whic* *h classifies such extensions up to equivalence [2,proposition 5.1]. We use this group and inducti* *on on the dimension of C to give a complete list of the cocommutative connected k-Hopf algebras of vec* *tor space dimension less than or equal to p3. A list of the commutative connected k-Hopf algebras o* *f dimension less than or equal to p3 could also be obtained by taking duals. In practice it can be difficult to compute Singer's cohomology group, or even* * to calculate the Hopf algebra structure on C determined by an element in that group. Fortunately, we * *can avoid these computations for all but a few specific cases. When C is commutative and k is a* * perfect field, the algebra structure is determined by Borel's theorem [1,theorem 7.11], and the po* *ssible coproducts are easily deduced when the dimension of C is small. If C is noncommutative but pri* *mitively generated, then it is the universal enveloping algebra of a nonabelian connected restricte* *d Lie algebra [1, theorem 6.11]. When the dimension of C is small, it is not difficult to give a list of * *these Lie algebras. We use Singer's theory for the remaining cases : C noncommutative and non-primitiv* *ely generated or C commutative and k not a perfect field. There are relatively few of these. Since Singer's theory is not widely known, we give a brief summary in section* * one. A deep understanding of his results is not necessary for our purposes, but we will use* * his terminology and his definition of the group classifying extensions. In section two we study central extensions A ! C ! B with A polynomial in one* * variable truncated at height p or exterior on one generator and with B and C cocommutati* *ve connected k-Hopf algebras. We are able to characterize those elements of H3(B; A) which d* *etermine extensions with C commutative or with C primitively generated. We also construct a small p* *iece of an exact sequence which is helpful in calculating H3(B; A). In section three we apply these results to low dimensional cocommutative conn* *ected k-Hopf algebras. We show that there are only two pairs (A; B) which can give a C of di* *mension less than or equal to p3 when C is noncommutative and non-primitively generated. We calcu* *late H3(B; A) in these cases and show that there are only two such Hopf algebras. Borel's the* *orem, supplemented by the theory of section two when k is not perfect, and restricted Lie algebras* * are used to complete the list. ____________ 1991 Mathematics Subject Classification. Primary 16W30 57T05 ; Secondary 18G6* *0. Typeset by AM * *S-TEX 1 2 GREGORY D. HENDERSON 1 Background Material. In this section we establish our notation and give an overview of Singer's th* *eory of extensions of connected Hopf algebras. We assume that the reader is familiar with the basic f* *acts about Hopf algebras to be found in Milnor and Moore [1]. The reader will find a knowledge * *of the classification of extensions of groups helpful in understanding Singer's theory, although such* * a knowledge is not necessary for this results in this paper. Notation. Let R be a commutative ring with unit. A Hopf algebra will mean a connected g* *raded R-Hopf algebra in the sense of Milnor and Moore [1,definition 4.1]. If C is such a Hop* *f algebra, we denote the product by C, the coproduct by C, the unit by jC, and the counit by fflC. * *The category of cocommutative connected R-Hopf algebras has the tensor product as a product [1,* *page 238] with projections ~= pA: A B 1fflB---!A R -! A and ~= pB: A B fflA1---!R B -! B: There are also two inclusions ~= 1jB iA: A -! A R ---! A B and ~= jA1 iB: B -! R B ---! A B: When this notation is ambiguous we will use numbers as subscripts to indicate t* *he factors in the tensor product to be used. Thus p13 : B B A ! B A indicates projection on th* *e first and third factors. The maps which permute (with appropriate signs) the terms in a tensor product* * will be given by a list of numbers in parentheses. For instance (1 3 2 4) : A1 A2 A3 A4 ! A1 A3 A2 A4: Finally, will be the free associative R-algebra, and R the free R-module, generated by the set {x1; . .;.xn}. Convolution products. The theory of extensions of groups makes use of the group structure on the se* *t of group homomor- phisms from G to H given by (f1+ f2)(g) = f1(g) + f2(g), f-1(g) = -f(g), and 0(* *g) = 0. With this product homgp(G; H) is an abelian group if H is commutative. The theory of exte* *nsions of connected Hopf algebras uses the group structure on the set of connected R-module morphis* *ms from B to A given by the convolution product[1,definition 8.1] : for f; g 2 hom(B; A), f * * *g = A(f g) B . The unit is the map jAfflB (the trivial morphism) and hom(B; A) is an abelian g* *roup when B is cocommutative and A is commutative. A connected R-module morphism is a R-module* * morphism which preserves the unit, i.e. f(1) = 1. The properties of the convolution product are easily established and seem to * *be common knowl- edge, so we will use the basic properties without proof. Singer writes the con* *volution product additively, but we have chosen to write it multiplicatively to avoid confusion * *with the product coming from the module structure. Singer's theory of extensions of connected Hopf algebras. Since Singer's theory does not appear to be widely known, we give a brief ove* *rview here. In actual fact we will only use his description of the group H3(B; A), which class* *ifies extensions of B by A, and of how elements of that group correspond to extensions. His theory is* * essentially a self dual version of the classical extension theory for groups. An extension of Hopf algebras is a sequence of the form A ! C ! B = C==A, whe* *re A is a left normal sub-Hopf algebra of C, together with a R-module splitting map fl : B ! C* *. Singer gives the following definition, which is equivalent by [1,proposition 4.9] : LOW DIMENSIONAL COCOMMUTATIVE CONNECTED HOPF ALGEBRAS 3 Definition 1.1 [2,definition 2.1]. An extension of a connected R-Hopf algebra B* * by a connected R-Hopf algebra A is a diagram of connected Hopf algebra morphisms A ff-!C -fi!B together with a map : C ! A B which is both an isomorphism of left A-modules * *and an isomorphism of right B-comodules. C is a left A-module via the map ff and the product on C and a right B-comodule* * via the coproduct on C and the map fi. Each extension determines an action oeA of B on A and a coaction aeB of A on * *B[2,definition 2.2 and proposition 2.3]. As in the group theory case the action of b on a is c* *omputed by lifting b to C by the splitting map fl : B ! C, performing the Hopf algebra analogue of c* *onjugation of ff(a) by fl(b), and pulling back to A. Thus ffoeA = (flpB ) * (ffpA) * (flpB )-1. The* * coaction is dual to this and can be thought of as "co-conjugation". The proofs of the following proposit* *ions are a useful exercise and are left to the reader : Proposition 1.2. If A is central in C, then the action oeA : B A ! A is the pr* *ojection pA (the trivial action). Proposition 1.3. If C is cocommutative, then the coaction aeB : B ! B A is the* * inclusion iB (the trivial coaction). The module isomorphism : C ! A B induces a product and a coproduct on A B.* * As in the group theory case the product can be described in terms of the action of B on A* * and a twisting function oA : B B ! A which measures the failure of the splitting map fl : B !* * C to preserve the product : ffoA = (flp1) * (flp2) * (fl)-1. Dually, the coproduct can be des* *cribed in terms of the coaction of A on B and a cotwisting function OEB : B ! A A. is an equivalence* * between the given extension and A iA-!A B pB--!B when A B has this product and coproduct. We will work exclusively with central extensions A ! C ! B which have A commu* *tative and both C and B cocommutative. In this case the action and coaction are trivial by* * lemmas 1.2 and 1.3, and so the product and coproduct are given by X (a b a b) = aaoA(b(1); b(1)) b(2)b(2) and (1.* *1) X (1) (2) (a b)= a(1)OEB (b(1)) b(2) a(2)OEB (b(1)) b(3); (1.* *2) P (1) (2) where OEB (b) = OEB (b) OEB (b) [2, propositions 2.4 and 2:4*]. The classification of extensions thus reduces to the classification of pairs * *(oA; OEB ). Singer describes a cohomology theory associated to a pair of Hopf algebras (A; B) where A is com* *mutative, B is cocommutative, A is a B-module, B is a A-comodule, and there are various condit* *ions on the action and coaction. He calls such a pairs abelian matched pairs [2,definition 3.1]. W* *hen the action and coaction are trivial, the extra conditions hold trivially and we have the trivi* *al abelian matched pair (A; B) for any A commutative and B cocommutative. The cohomology of a group G, H*(G; A), can be computed using the normalized b* *ar resolution of k as a module over the group ring of G and applying homgp(-; A) to get a cha* *in complex. Singer makes this self dual by taking an analogue of the normalized bar resolution of * *k as a B-module and an analogue of the normalized cobar resolution of k as an A-comodule and applyi* *ng an appropriate 4 GREGORY D. HENDERSON hom to get a bicomplex [2,pages 11-12] : 0 x? x ? dv?? 0 ----! hom(B; A A) --dh--!hom(B B; A A) x? x x ? dv?? dv?? 0 ----! hom (B; A) --dh--! hom (B B; A) --dh--! x? x x ? ?? ?? 0 ----! 0 ----! 0 ----! 0 His cohomology, H*(B; A), comes from the total complex of this bicomplex. Thus * *the 3-cocycles are pairs (oA; OEB ) for oA : B B ! A and OEB : B ! A A satisfying five condition* *s [2,proposition 4.1.5], given here for the trivial abelian matched pair : The unit condition (from normalization in the bar resolution) oAi1 = jAfflB = oAi2: (1.* *3) The counit condition (from normalization in the cobar resolution) p1OEB = jAfflB = p2OEB : (1.* *4) The associativity condition (from dhoA trivial) (oAp12) * (oA(B 1)) = (oAp23) * (oA(1 B )): (1.* *5) The coassociativity condition (from dvOEB trivial) ((1 A)OEB ) * (i23OEB ) = (( A 1)OEB ) * (i12OEB ): (1.* *6) The Hopf condition (from dvoA = (dhOEB )-1) ( AoA) * (OEB B ) = (AA (OEB OEB )) * ((oA oA) BB ): (1.* *7) In other words, the 3-cocycles are pairs of twisting and cotwisting functions (* *oA,OEB ) defining a product and coproduct on A B by (1.1)and (1.2)which satisfy the conditions for* * a Hopf algebra. Note that, when written additively, dh and dv resemble the differentials in t* *he bar and cobar constructions, but they use the convolution product, and so are actually quite * *different. The 3-coboundaries are pairs (o ; OE ) defined by some : B ! A by [2,proposi* *tion 4.1.6], given here for the trivial abelian matched pair : o = (p1) * (-1B ) * (p2) (1.* *8) OE = (i1-1) * ( ) * (i2-1): (1.* *9) The 3-coboundaries give the extensions which are equivalent to the extension de* *fined by a triv- ial twisting and cotwisting function (the trivial extension. Thus 3-cocycles d* *iffering by a 3- coboundary define equivalent extensions and H3(B; A) classifies extensions of B* * by A up to equiva- lence [2, proposition 5.1]. LOW DIMENSIONAL COCOMMUTATIVE CONNECTED HOPF ALGEBRAS 5 2 Extensions Associated with Cocommutative Connected Hopf Algebras. In this section A will denote a polynomial algebra on one generator truncated* * at height p or an exterior algebra on one generator. x will be the generator of A and ht(x) will * *be the height of x in A (either p or 2 respectively). Furthermore, B will denote a cocommutative connec* *ted k-Hopf algebra and all abelian matched pairs will be trivial. We begin by showing that every f* *inite dimensional cocommutative connected k-Hopf algebra occurs in an extension A ! C ! B corresp* *onding to an element in H3(B; A). Conversely, if ht(x) = p, then every element of H3(B; A* *) determines an extension with C cocommutative. If ht(x) = 2 < p, then the elements that determ* *ine extensions with C cocommutative are precisely those which have a representative with a tri* *vial cotwisting function. Next we give necessary and sufficient conditions for an element in H3(B; A) t* *o determine an extension with C commutative or with C primitively generated. Lastly we constr* *uct an exact sequence which will be used to compute H3(B; A). Extensions with C cocommutative. Cocommutative connected k-Hopf algebras for k a field of positive characteris* *tic p have many properties which are analogous to those of p-groups. The following lemma is an* *alogous to the existence of nontrivial central elements in a p-group and shows that there is a* * simple type of extension which can be used to study cocommutative connected k-Hopf algebras. Proposition 2.1. If C is a nontrivial finite dimensional cocommutative connecte* *d k-Hopf algebra for k a field of positive characteristic p, then there is an extension A ! C ! * *B determining an element in H3(B; A) where B is a cocommutative connected k-Hopf algebra, A is e* *ither k[x]=(xp) with |x|even if p > 2 or k(x) with |x|odd if p > 2, and (A; B) is the trivial a* *belian matched pair. Proof. Such a C always contains a primitive central element (see for example [3* *, proposition 1.2]). Thus there is always a central sub-Hopf algebra A of the stated form and a cent* *ral extension A ! C ! B = C==A. The action of B on A is trivial by proposition 1.2 since A is* * central and the coaction of A on B is trivial by proposition 1.3 since C is cocommutative. In general, not all elements in H3(B; A) determine extensions with C cocommut* *ative. However, for the extensions in proposition 2.1 it is possible to completely characterize* * those elements which do. This provides a way to generate all the cocommutative connected k-Hopf alge* *bras inductively using H3(B; A). Theorem 2.2. Consider an extension A ! C ! B corresponding to an element u 2 H3* *(B; A) where B is a cocommutative connected k-Hopf algebra and (A; B) is the trivial m* *atched pair. (a)If A = k[x]=(xp) with |x|even if p > 2, then C is cocommutative. (b)If A = k(x) with |x|odd and p > 2, then C is cocommutative if and only if* * there is a representative for u with trivial cotwisting function. Proof. The expression for the coproduct on C ~=A B (1.2)shows that C is cocomm* *utative if and only if the cotwisting function for a representative of u is symmetric (in the * *graded sense). The theorem follows from the following lemma, which ensures the existence of a repr* *esentative for u with a particularly simple cotwisting function. Lemma 2.3. Assume that B is a cocommutative connected k-Hopf algebra for k a fi* *eld of positive characteristic p, that A = k[x]=(xp) or k(x), and that (A; B) is the trivial ab* *elian matched pair. If (oA; OEB ) is a cocycle representing an element of H3(B; A), then (oA; OEB )* * is cohomologous to a cocycle (o0A; OE0B) with 8 >>>b 0 if1|b|= 0 >>< P 1 C OE0B(b) = > fl(b) B@ ___rx!sr! xsAif |b|= ht(x) |x| (2.* *1) >>> r+s=ht(x) >: r;s>0 0 otherwise, 6 GREGORY D. HENDERSON Here fl : Bht(x)|x|! k is some k-linear function. Furthermore, (o0A; OE0B) can* * be chosen so that o0A= oA in degrees less than 2 |x|. Proof. A A is concentrated in degrees which are multiples of |x|. Since OEB :* * B ! A A is a connected module morphism which satisfies the counit condition(1.4), it must be* * trivial in degree less than 2 |x|. Now, if (o00A; OE00B) is a cocycle such that OE00Bis trivial * *in degrees less than l |x|for some 2 l < ht(x), then the coassociativity condition (1.6)in degree l |x|becom* *es (1 A)OE00B= ( A 1)OE00B. A simple calculation shows that OE00B= fl0 A(xl) in that degree fo* *r some k-linear function fl0: Bl|x|! k. Adding the coboundary (o ; OE ) defined by 8 > -fl0(b)xlif |b|= l |x| : 0 otherwise gives a cocycle (o0A; OE0B) cohomologous to (o00A; OE00B) with o0A= o00Ain degr* *ees less than 2 |x|(since l 2) and OE0Btrivial in degrees less than (l + 1) |x|. Inductively we obtain * *a cocycle (o0A; OE0B) cohomologous to (oA; OEB ) with o0A= oA in degrees less than 2 |x|and OE0Btrivi* *al in degrees less than ht(x) |x|. We use the coassociativity condition (1.6)again to conclude that OE0Bsatisfie* *s (2.1)in degree ht(x) |x|and that it is trivial in all higher degrees. Thus if ht(x) = p, there is a representative for u with a cotwisting function* * which is given by (2.1). If p > 2, then |x|is even, so the cotwisting function is symmetric and C* * is cocommutative. If p = 2, then signs are irrelavent, so the cotwisting function is symmetric and C* * cocommutative. On the other hand, if ht(x) = 2 < p, then |x|is odd and the cotwisting function ha* *s the form fl . (x x), which is symmetric only if fl is zero. Hence C is cocommutative if and only if * *u has a representative with a trivial cotwisting function. Note that the proof of lemma 2.3 is essentially a calculation of Cotor1A(B; k* *), where B is a trivial A-comodule, using the cobar construction on A. The difference is that l* *emma 2.3 uses the convolution product instead of the usual sum of module morphisms. Extensions with C commutative. Consider a central extension A ! C ! B. As in the classification of extension* *s of groups, we think of the product in C as the products on A and B plus the value of oA. This* * suggests that C is commutative if and only if B is commutative and oA is symmetric (in the graded * *sense). Actually, we need to check the symmetry of oA only on the irreducibles of B, and when A i* *s k[x]=(xp) or k(x) we need to check only in degree |x|. The following theorem makes this prec* *ise. Theorem 2.4. Assume that B is a cocommutative connected k-Hopf algebra for k a * *field of positive characteristic p, that A is either k[x]=(xp) with |x|even if p > 2 or k(x) with* * |x|odd if p > 2, and that (A; B) is the trivial abelian matched pair. If u 2 H3(B; A) determines an * *extension A ! C ! B and (oA; OEB ) is a cocycle representing u, then C is commutative if and only i* *f B is commutative and oA is trivial on (QB ^ QB)|x|. Proof. It is certainly a necessary condition that B be commutative, so we assum* *e that this is the case. The expression for the product on C ~=A B (1.1)shows that C is commutati* *ve if and only if the twisting function is symmetric for any representative of u. Before proceeding, we need to show that oA gives a well defined map on (QB ^ * *QB)|x|. Since oA : BB ! A, and A is trivial in degrees below |x|, the associativity condition(1.5)* *on (IBIBIB)|x| is oA(B 1) = oA(1 B ). Thus oA(bc; d) = oA(b; cd) for b c d 2 (IB IB IB)|* *x|. Since B is commutative, this implies that oA(bc; d) = (-1)|bc||d|oA(d; bc), and therefo* *re oA is well defined on (QB ^ QB)|x|. If C is commutative, then oA is symmetric, and so is zero on (QB ^ QB)|x|. Co* *nversely, if oA is zero on (QB ^ QB)|x|, we will show that oA is symmetric, and so C is commutativ* *e. Assume that oA is zero on (QB ^ QB)|x|. By the unit condition(1.3), it suffices to show tha* *t oA is symmetric on IB IB. Since A is concentrated in degrees l |x|for 0 l < ht(x), we only ne* *ed to consider LOW DIMENSIONAL COCOMMUTATIVE CONNECTED HOPF ALGEBRAS 7 (IB IB)l|x|. The case l = 0 is trivial. For l = 1, the initial assumption on o* *A and the fact that oA(bc; d) = (-1)|bc||d|oA(d; bc) for b c d 2 (IB IB IB)|x|imply that oA is * *symmetric on all of (IB IB)|x|. To finish the proof, we argue by induction on l using the Hopf condition (1.7* *). To simplify the calculation, we assume that OEB is trivial in degrees less than ht(x) |x|. By l* *emma 2.3, this can be done without loss of generality. Assume then that oA is symmetric in degrees le* *ss than l |x|for some 1 < l < ht(x) and take b b2 (IB IB)l|x|. The Hopf condition on b bis X AoA(b; b) = oA(b(1); b(1)) oA(b(2); b(2)): By assumption, the only terms on the right which are possibly non-symmetric in * *b and bare oA(b; b) 1 and 1 oA(b; b). Thus the Hopf condition implies that AoA(b; b) is symmetric* * in b and b. But oA(b; b) is a multiple of xl in A, and since l > 1 it follows that oA(b; b) is * *symmetric. Hence oA is symmetric on (IB IB)l|x|, and we are done. Extensions with C primitively generated. Intuitively, if C occurs in an extension A ! C ! B with A primitively generat* *ed, then C is primitively generated if and only if there is a set of algebra generators in B * *which are primitive, and which remain primitive when lifted to C. The first part of this condition is th* *at B be primitively generated and the second part is controlled by the behavior of the cotwisting f* *unction on primitive irreducibles. This is made precise in the following theorem. Theorem 2.5. Assume that B is a finite dimensional cocommutative connected k-Ho* *pf algebra for k a field of positive characteristic p, that A is primitively generated, and th* *at (A; B) is the trivial abelian matched pair. If u 2 H3(B; A) determines an extension A ff-!C -fi!B; then C is primitively generated if and only if B is primitively generated and t* *here is a representative for u with a cotwisting which is zero on the primitive irreducibles of B. Proof. The following diagram is commutative with exact rows [1, proposition 4.1* *0]. 0 ! P A ! P C ! P B # # # (2.* *2) QA ! QC ! QB ! 0: Assume first that B is primitively generated and that (oA; OEB ) is a cocycle r* *epresenting u with OEB zero on the primitive irreducibles of B. Since B is primitively generated, we c* *an chose a basis {bi} for QB such that bi is primitive for all i. Since A is primitively generated, w* *e can chose a basis {ai} [ {a0i} for A such that aiand a0iare primitive and the kernel of QA ! QC i* *s spanned by {a0i}. Thus QC has a basis {ai 1} [ {1 bi}. The formula for the coproduct on C ~=A B* * (1.2)shows that 1 biand ai 1 are primitive, and so C is primitively generated. Conversely, if C is primitively generated, then so is B as a quotient of C. B* *y [1, proposition 6.16], P C ~=P A P B. Thus C has a vector space basis {ff(X)IY J} where X = (x1; . .;* *.xn) for {xi} a basis for P A, Y = (y1; . .;.yl) for yiprimitive in C and {fi(yi)} a basis for * *P B, and I = (i1; . .;.in) and J = (j1; . .;.jl) for 0 ik < 2 if |xk|odd, 0 ik < p if |xk|even, 0 jk < * *2 is |yk|odd, and 0 jk < p if |yk|even. The map : C ! A B given by (ff(X)IY J) = XI fi(Y )J i* *s clearly an isomorphism of left A-modules and right B-comodules, and thus determines a r* *epresentative (oA; OEB ) for u [2, proposition 2.11]. OEB is given by definition 2.2 and pro* *position 2.3 of [2], and is easily seen to give zero on {fi(yi)} (since yi primitive and pA(yi) = 0), an* *d hence on all the primitives of B. An exact sequence. Lemma 2.3 says that the cotwisting function for an element in H3(B; A) is det* *ermined by a k- linear map fl : Bht(x)|x|! k. The dual statement is that the twisting function * *is determined by a k-linear map fl0: (IB IB)|x|! k, which is why the symmetry of oA only needs to* * be checked in degree |x|in theorem 2.4. Of course, the fl and fl0 which come from representat* *ives of elements in H 3(B; A) are not arbitrary, and different representatives give different maps.* * The following theorem develops this idea. 8 GREGORY D. HENDERSON Theorem 2.6. Assume that B is a cocommutative connected k-Hopf algebra for k a * *field of positive characteristic p, that A is either k[x]=(xp) with |x|even if p > 2 or k(x) with* * |x|odd if p > 2, and that (A; B) is the trivial matched pair. There is an exact sequence of abelian * *groups H1;ht(x)|x|(B; k) ! H3(B; A) ! H2;|x|(B; k): This is actually part of a longer exact sequence which will de described else* *where. H *;*(B; k) is the ordinary cohomology of a Hopf algebra, i.e. Ext*;*B(k; k). Proof. H *;*(B; k) is the cohomology of the normalized bar construction on B : ! IB IB IB -1-1-----!IB IB -! IB -0!k ! 0; where IBn is in external degree n. Thus H1;ht(x)|x|(B;=k){fl : IBht(x)|x|! k | fl = 0} {fl0: (IB IB)|x|! k | fl0( 1) = fl0(1 )} H2;|x|(B; k)= ______________________________________{ | : IB * *|x|! k} Consider : H1;ht(x)|x|(B; k) ! H3(B; A) induced by !(fl) = (o0A; OE0B) where* * o0Ais trivial and OE0Bis given by (2.1). If fl is a cocycle, we claim that (o0A; OE0B) is a cocyc* *le. The unit, counit, and associativity conditions ((1.3),(1.4), and (1.5)) hold trivially, and the coass* *ociativity condition (1.6) holds by the same calculation done in the proof of lemma 2.3. Since o0Ais trivi* *al, the Hopf condition (1.7)is OE0BB = AA (OE0BOE0B). The left hand side is zero on IB IB since fl =* * 0, while the right hand side is zero on IB IB since x has height ht(x). On the remaining part of * *B B both sides give OE0B. Note that ! also takes coboundaries to coboundaries since !(0) has t* *rivial twisting and cotwisting functions. Finally, if !(fl1) = (oA; OEB ) and !(fl2) = (o0A; OE0B),* * then the twisting function for !(fl1+ fl2) is oA * o0Asince all are trivial. The cotwisting function for !* *(fl1+ fl2) is OEB * OE0Bsince both OEB and OE0Bare trivial in degrees less than ht(x) |x|, so in that degree * *OEB *OE0B= OEB +OE0B. Thus ! induces a group homomorphism on cohomology. Next consider : H3(B; A) ! H2;|x|(B; k) induced by (oA; OEB ) = fl0, where o* *A(b; b) = fl0(b; b)x for all b b2 (IB IB)|x|. If (oA; OEB ) is a cocycle, then the associativity c* *ondition(1.5)in degree |x|is oA(B 1) = oA(1 B ), which implies that fl0 is a cocycle. If (oA; OEB )* * = (o ; OE ) is the coboundary defined by : B ! A, we consider : IB|x|! k such that (b) = (b)x fo* *r b 2 IB|x|. The definition of o (1.8)gives o (b; b) = -(bb) for b b2 (IB IB)|x|and so fl* *0is a coboundary since fl0= - . To see that is additive, simply note that oA * o0A= oA + o0Ain * *degree |x|. It remains to show that the sequence is exact. Clearly ! is zero, since !(fl* *) has a trivial twisting function. Conversely, if (oA; OEB ) is a cocycle such that (oA; OEB )* * is a coboundary, then oA(b; b) = fl0(bb)x for b b2 (IB IB)|x|for some fl0: IB|x|! k. Consider : B * *! A given by 8 > fl0(b)xif |b|= |x| : 0 otherwise If o is the twisting function of the coboundary defined by (1.8), then (oA * * *o )(b; b) = oA(b; b) - (bb) = 0 on (IB IB)|x|. Thus (oA; OEB ) is cohomologous to a (o0A; OE0B) with o* *0Atrivial in degree |x|. By lemma 2.3, we can assume that OE0Bis given by (2.1)for some fl. We claim tha* *t o0Ais trivial in all degrees. Assume that o0Ais trivial in degrees less than l |x|for some 1 < l < h* *t(x) and consider the Hopf condition (1.7)on b b2 (IB IB)|x|. Since OE0Bis trivial in this degree a* *nd below, the Hopf condition is Ao0A(b; b) = 0. But o0A(b; b) is a multiple of xl, and so is prim* *itive only if it is zero. Thus o0Ais trivial and (o0A; OE0B) = !(fl). fl is a cocycle since the Hopf cond* *ition on (IB IB)ht(x)|x| is OE0BB = 0 when the twisting function is trivial and the cotwisting function * *is given by (2.1). LOW DIMENSIONAL COCOMMUTATIVE CONNECTED HOPF ALGEBRAS 9 3 The Classification of Low Dimensional Cocommutative Connected k-Hopf Algebras. In this section we apply the results of section two to obtain a list of the c* *ocommutative connected k-Hopf algebras of vector space dimension less than or equal to p3 when k is a * *field of positive characteristic p. We begin by using Borel's theorem to list the ones which are * *commutative when k is perfect, and restricted Lie algebras to list those which are noncommutativ* *e but primitively generated. We then use the results of section two to show that there are only * *two which are noncommutative and non-primitively generated. Finally, we indicate how the theo* *ry can be used to account for fields which are not perfect. To simplify the terminology, we note that proposition 2.1 implies that the ve* *ctor space dimension of a finite dimensional cocommutative connected k-Hopf algebra is 2npm . This i* *s also a consequence of Borel's theorem [1, proposition 7.11] if k is a perfect field. We call n + m* * the total rank of the Hopf algebra. The commutative cases. Lemma 3.1. There are two commutative cocommutative connected k-Hopf algebras of* * total rank one over a field k of positive characteristic p : (a)k(x) with p > 2 and |x|odd, and (b)k[x]=(xp) for |x|even if p > 2. All algebra generators are primitive. Type (a) has dimension 2 and type (b) has* * dimension p. Proof. This is a direct consequence of [1, proposition 7.8] Theorem 3.2. There are five commutative cocommutative connected k-Hopf algebras* * of total rank two over a perfect field k of positive characteristic p : (a)k(y; z) for p > 2 and |y|and |z|odd; (b)k(y) k[z]=(zp) for p > 2, |y|odd, and |z|even; (c)k[y]=(yp2) with |y|even if p > 2; (d)k[y; z]=(yp; zp) with |y|and |z|even if p > 2; and (e)k[y; z]=(yp; zp) with |z|= p |y|, |y|even if p > 2, and coproduct (z) = X __1_yr ys: r+s=pr ! s ! r;s>0 Unless otherwise stated, all algebra generators are primitive. Type (a) has dim* *ension 4, type (b) has dimension 2p, and types (c),(d), and (e) have dimension p2. Proof. Proposition 7.21 of [1] implies that for p > 2, k(x; y) with |x|and |y|o* *dd is the only Hopf algebra of dimension 4. By the same reasoning, (x) k[y]=(yp) with |x|odd and |* *y|even is the only Hopf algebra of dimension 2p for p > 2. By Borel's theorem [1,proposition 7.11], there are two possible alegbra struc* *tures for a Hopf algebra of dimension p2 : k[y]=(yp2) and k[y; z]=(yp; zp), where |y|and |z|are * *even if p > 2. For the first there is only one possible Hopf algebra structure, while for the second, * *we may assume that |y| |z|, and so y is primitive. If z is primitive, the Hopf algebra structure i* *s clear. If not, consider the dual. Since QC* ~=(P C)*, we know that C* is monogenic and so C* = k[w]=(wp* *2). A simple calculation shows that the coproduct on z has the form stated in the Hopf algeb* *ra of type (e). Theorem 3.3. There are twelve commutative cocommutative connected k-Hopf algebr* *as of total rank three over a perfect field k of positive characteristic p : type (a) (p > 2, dimension 8) (x; y; z) |x|,|y|, and |z|odd; type (b) (p > 2, dimension 4p) 10 GREGORY D. HENDERSON (x; y) k[z]=(zp) |x|and |y|odd, |z|even; type (c) (p > 2, dimension 2p2) 2 (x) k[y]=(yp ) |x|odd and |y|even; type (d) (p > 2, dimension 2p2) (x) k[y; z]=(yp; zp)|x|odd, |y|and |z|even; type (e) (p > 2, dimension 2p2) (x) k[y; z]=(yp; zp)|x|odd, |y|even, |z|= p |y|, and (z) = X __1_yr ys; r+s=pr ! s ! r;s>0 type (f) (dimension p3) 3 k[x]=(xp ) |x|even if p >;2 type (g) (dimension p3) 2 k[x; y]=(xp; yp )|x|and |y|even if p > 2 type (g-1) (x) = 0 (y) = 0; type (g-2) |y|= p |x| (x) = 0 (y) = X __1_xr xs; r+s=pr ! s ! r;s>0 type (g-3) |x|= p2|y| (x) = X __1_yrp ysp r+s=pr ! s ! r;s>0 (y) = 0; type (h) (dimension p3) k[x; y; z]=(xp; yp; zp)|x|, |y|, and |z|even if p > 2 LOW DIMENSIONAL COCOMMUTATIVE CONNECTED HOPF ALGEBRAS 11 type (h-1) (x) = 0 (y) = 0 (z) = 0; type (h-2) |z|= p |y| (x) = 0 (y) = 0 (z) = X __1_yr ys; r+s=pr ! s ! r;s>0 type (h-3) |y|= p |x| |z|= p2|x| (x) = 0 (y) = X __1_xr xs r+s=pr ! s ! r;s>0 (z) = X __1_yr ys - (y) (y)p-1: r+s=pr ! s ! r;s>0 Unless otherwise mentioned, all algebra generators are primitive. Proof. For p > 2, the Hopf algebras with dimensions other than p3 (types (a) th* *rough (e)) are given by proposition 7.21 of [1] and theorem 3.2. The algebra structures of the remai* *ning Hopf algebras are given by Borel's theorem [1, theorem 7.11]. There is only one possible Hopf* * algebra structure on type (f). The Hopf algebras of type (g-1) and (h-1) are the primitively gene* *rated cases, and we proceed to classify the non-primitively generated commutative Hopf algebras wit* *h QC either two or three dimensional. 2 Assume that C is non-primitively generated and C = k[x; y]=(xp; yp ) (thus |y* *|6= |x|). If |x|< |y|, then x is primitive, but [y] 2 QC cannot be represented by a primitive element.* * A calculation with the coassociativity condition on C which is very similar to the calculation do* *ne for OEB in lemma 2.3 shows that type (g-2) is the only possibility. If |y|< |x|, then y is primi* *tive and [x] 2 QC cannot be represented by a primitive element. If |x|= p |y|, then coassociativity gives (x) = X __1_yr ys: r+s=pr ! s ! r;s>0 But with this coproduct is not an algebra map since (xp) 6= (x)p. On the o* *ther hand, if |x|6= p |y|, we consider the dual of C. P C* = and QC* = , where x* * is dual to x, y is dual to y, and zis dual to yp. Since ypmust be primitive but is linearly indepe* *ndent of x and y, it must be zero, and this together with the relative degrees of yand zshows that C* ** is of type (g-2). A simple calculation shows that the dual of type (g-2) is of type (g-4). Lastly, assume that C is non-primitively generated and C = k[x; y; z]=(xp; yp* *; zp). If x and y are primitive but z is not, then QC* = and P C* = , and so* * C* is primitively generated with two algebra generators. Direct calculation shows that the dual o* *f type (g-1) is of type (h-2). If x is primitive but y and z are not, then QC* = , so C* is of * *type (f). Again, direct calculation shows that the dual of type (f) is of type (h-3). The noncommutative primitively generated cases. 12 GREGORY D. HENDERSON Lemma 3.4. There are no noncommutative primitively generated k-Hopf algebras of* * total rank one for k a field of positive characteristic p. Proof. If C is such a Hopf algebra, then [1, theorem 6.11] implies that C is th* *e universal enveloping algebra of a nonabelian connected restricted Lie algebra of dimension one. But * *a one dimensional Lie algebra must be abelian, so this is impossible. Theorem 3.5. There is only one noncommutative primitively generated connected H* *opf algebra of total rank two over a field k of positive characteristic p : =(y2p); where the brackets denote the free associative k-algebra generated by y, |y|is * *odd, and p > 2. Proof. The noncommutative primitively generated connected Hopf algebras of tota* *l rank two are the enveloping algebras of the nonabelian connected restricted Lie algebras of dime* *nsion two[1, theorem 6.11]. Assume that L is such a restricted Lie algebra. The first case to cons* *ider is p > 2 and Lodd= k. Since the commutator of odd degree elements is in even degrees, * *this Lie algebra must be abelian. The next case is p > 2, Lodd= k, and Leven= k, and here * *there is only one possible nontrivial commutator, which by proper choice of generators is [y; y] * *= z. The restriction map must be trivial by degree considerations, and this gives the Hopf algebra l* *isted in the statement of the theorem. The last case is p > 2, Lodd= 0, and Leven= k or p = 2 an* *d L = k. Here the commutator is forced to be zero and so L is abelian. Theorem 3.6. There are twelve noncommutative primitively generated connected Ho* *pf algebras of total rank three over a field k of positive characteristic p : type (a) (p > 2, dimension 4p) =(y2p) k(z) |y|and |z|odd; type (b) (p > 2, dimension 4p) =(y2p; z2 - ly2; [y;lz])nonzero in k |z|= |y| |y|odd; type (c) (p > 2, dimension 4p) =(y2p; z2; [y; z] -|y2)z|= |y| |y|odd; type (d) (p > 2, dimension 4p) =(y2p; z2 - ly2; [y; z]l-ny2)onzero in k |z|= |y| |y|odd; type (e) (p > 2, dimension 4p) =(xp; y2; z2; [x; y]; [x; z];|[y;xz]|-=x)|y|+ |z| |y|and |z|odd ; LOW DIMENSIONAL COCOMMUTATIVE CONNECTED HOPF ALGEBRAS 13 type (f) (p > 2, dimension 4p) =(x2; y2; zp; [x; y]; [x; z];|[y;xz]|-=x)|y|+ |z| |y|odd and |z|even ; type (g) (p > 2, dimension 2p2) 2 =(y2p ) |y|odd ; type (h) (p > 2, dimension 2p2) =(y2p) k[z]=(zp) |y|odd and |z|even ; type (i) (p > 2, dimension 2p2) =(y2p; zp - ly2; [y;lz])nonzero in k p |z|= 2 |y| |y|odd and |z|even ; type (j) (dimension p3) =(xp; yp; zp; [x; y]; [x; z];|[y;xz]|-=x)|y|+ |z| |y|and |z|even if p > 2; type (k) (dimension p3) 2 p p =(yp ; z ; [y; z] -|yz)|= (p - 1) |y|; |y|even if p > 2. type (l) (p = 2, dimension 8) =(y4; z4; y2 - z2; [y; z]|-zy2)|= |y|; Again, the brackets denote the free associative k-algebra generated by the give* *n elements. If p = 2 and |y|= |z|, then types (j) and (k) coincide. Depending on the field k, some o* *f the Hopf algebras in case (b) or case (i) may coincide for certain values of l. Proof. As before, the noncommutative primitively generated connected Hopf algeb* *ras of total rank three are the enveloping algebras of the nonabelian connected restricted Lie al* *gebras of dimension three[1, theorem 6.11]. Assume that L is such a restricted Lie algebra. The fir* *st case to consider is 14 GREGORY D. HENDERSON p > 2 and Lodd= k. In this case it is impossible for L to be nonabelia* *n since the commutator of odd degree elements lies in even degree. The next case is p > 2, Lodd= k, and Leven= k. Degree and scaling co* *nsiderations give the following combinations of nonzero commutators to be considered : case 1 - [y; y] = x (|x|= 2 |y|) ; case 2 - [y; y] = x and [z; z] = lx for l nonzero in k (|x|= 2 |y|and |z|= |y|)* * ; case 3 - [y; y] = x and [y; z] = x (|x|= 2 |y|and |z|= |y|) ; case 4 - [y; y] = x, [z; z] = lx, and [y; z] = x for l nonzero in k (|x|= 2 |y|* *and |z|= |y|) ; case 5 - [y; z] = x (|x|= |y|+ |z|) ; and case 6 - [x; z] = y (|y|= |x|+ |z|). Because of the degrees the restriction map on x must be trivial. This gives the* * Hopf algebras of types (a), (b), (c), (d), (e), and (f), where we have relabeled the generators * *in (f) so that x is the central element. The third case is p > 2, Lodd= k, and Leven= k. In this case the deg* *rees of x, y, and z allow only one nontrivial commutator, and without loss of generality we have [x* *; x] = y. Considering the restriction maps gives three cases : case 1 - [x; x] = y, (y) = z, and (z) = 0 (|y|= 2 |x|and |z|= p |y|) ; case 2 - [x; x] = y, (y) = 0, and (z) = 0 (|y|= 2 |x|). case 3 - [x; x] = y, (y) = 0, and (z) = ly for l nonzero in k (|y|= 2 |x|, and * *p |z|= 2 |x|). These give the Hopf algebras of type (g), (h), and (i) respectively, and as bef* *ore, we have relabeled the generators. The fourth and last case is for p > 2, Lodd trivial, and Leven= k or* * p = 2 and L = k. Without loss of generality, we can assume that [y; z] = x, and the * *degrees then imply that [x; y] = [x; z] = 0 and (x) = 0. Again, degree and scaling considerations give * *the following nonzero commutators and restriction maps : case 1 - [y; z] = x (|x|= |y|+ |z|) ; case 2 - [y; z] = x and (y) = x (|x|= p |y|and |z|= (p - 1) |y|) ; case 3 - [y; z] = x, (y) = x, and (z) = x (p = 2, |x|= 2 |y|, and |z|= |y|) . These give the Hopf algebras of types (j), (k), and (l) respectively. To see that type (j) and (k) are isomorphic when p = 2 and |y|= |z|, consider* * the map from type (k) with generators y and z to type (j) with generators x, y, and zgiven b* *y fl(y) = y+ zand fl(z) = y. It is easily checked that this is an isomorphism of Hopf algebras. The noncommutative non-primitively generated cases. Lemma 3.7. There are no noncommutative non-primitively generated cocommutative * *connected k-Hopf algebras of total rank one over a field k of positive characteristic p. Proof. By lemma 2.1, this is clear. Theorem 3.8. There are no noncommutative non-primitively generated cocommutativ* *e connected k-Hopf algebras of total rank two over a field k of positive characteristic p. Proof. By lemma 2.1, a noncommutative non-primitively generated cocommutative H* *opf algebra C of total rank two occurs in an extension A ! C ! B with A = k[x]=(xp) or k(x) a* *nd B of rank one. Since all such B are commutative, theorem 2.4 implies that QB ^ QB must be* * nontrivial in degree |x|. This in turn implies that B = k(y) with |y|odd, p > 2, and |x|= 2 |* *y|. Thus |x|is even and so A = k[x]=(xp). Now, since B is primitively generated, theorem 2.5 and le* *mma 2.3 imply that QB2p|y|is nontrivial, but this is impossible since QB is concentrated in degree* * |y|. Theorem 3.9. There are two noncommutative non-primitively generated cocommutati* *ve connected k-Hopf algebras of total rank three over a perfect field k of positive characte* *ristic p : Type (a) ( p > 2, dimension 2p2 ) =(y2p) k[z]=(zp) |z|= 2p |y| |y|odd LOW DIMENSIONAL COCOMMUTATIVE CONNECTED HOPF ALGEBRAS 15 X (z) = __1_y2r y2s; r+s=pr ! s ! r;s>0 Type (b) ( dimension p3 ) =(xp; yp; zp - xp-1y; [x; y]; [x; z]; [y; z] - x) |x|= (p + 1) |y| |z|= p |y| |y|even if p > 2 (x) = (y) = 0 (z) = X __1_yr ys: r+s=pr ! s ! r;s>0 The brackets denote the free associative k-algebra generated by the given eleme* *nts, and generators are primitive unless otherwise noted. Proof. By lemma 2.1 a noncommutative non-primitively generated cocommutative co* *nnected k-Hopf algebra C of rank three occurs in an extension A ! C ! B with A = k[x]=(xp) or * *k(x) and B of rank two. Thus there2are six cases to consider. Case 1 : B = =(yp ) for p > 2 and |y|odd. B is primitively generated, so by theorem 2.5 and lemma 2.3 we must have QBht* *(x)|x|nontrivial. But this is impossible since ht(x) |x|is always even and QB is concentrated in * *odd degrees. Case 2 : B = k(y; z) for p > 2 and |y|; |z|odd. As in the previous case QBht(x)|x|must be nontrivial. But QB is concentrated * *in odd degrees, so this is impossible. Case 3 : B = k(y) k[z]=(zp) for p > 2, |y|odd, and |z|even. Again, QBht(x)|x|must be nontrivial, and so |z|= ht(x) |x|. Since B is commut* *ative, theorem 2.4 implies that (QB ^ QB)|x|= k|x|is nontrivial. Degree con* *siderations imply that |x|= 2 |y|, and so A = k[x]=(xp) and |z|= 2p |y|. The exact sequence from theorem 2.6 and lemma 2.3 in this case show that elem* *ents of H3(B; A) are represented by (oA; OEB ) with oA determined by oA(y; y) = ax and OEB trivi* *al except for 0 1 X 1 C OEB (z) = b B@ ____xr xsA : r+s=p r ! s ! r;s>0 Elements giving C noncommutative and non-primitively generated have a and b non* *-zero, and by rescaling x and z we can assume a = b = 1. (The extensions for the different (a* *; b) with a and b nonzero are distinct, but the C they determine are isomorphic). By (2.2), the algebra generators of C are y= 1y, z= 1z, and x= x1 if the late* *r is irreducible. A simple application of the formula for the product (1.1)shows that [x; y] = [x* *; z] = [y; z] = 0, the last since oA(y; z) is in dimension (2p + 1) |y|> p |x|, and so is zero. Since * *oA(y; y) = x, we have y2 = xand so {yizj} spans C. From Singer's formula for the coproduct (1.2)it is easy to see that yis primi* *tive and (z) = X __1_xr xs: r+s=pr ! s ! r;s>0 Since y is primitive, it generates a sub-Hopf algebra, and so the height of y m* *ust be 2p. As a consequence zpmust be a power of y. Degree considerations show that zp= 0 and t* *his gives the Hopf algebra of type (a). Case 4 : B = k[y]=(yp2) for |y|even if p > 2. 16 GREGORY D. HENDERSON B is commutative and QB ^ QB is trivial, which is impossible by theorem 2.4. case 5 : B = k[y; z]=(yp; zp) for |y|; |z|even if p > 2. B is commutative and QB ^ QB = k, so by theorem 2.4, we have |x|* *= |y|+ |z|and A = k[x]=(xp). B is primitively generated, so by theorem 2.5 and lemma 2.3 we m* *ust have QBp|x| nontrivial. But this is impossible since |y|; |z|< |x|. Case 6 : B = k[y; z]=(yp; zp) with (z) = X __1_yr ys r+s=pr ! s ! r;s>0 and |y|; |z|even if p > 2. As in case five we have |x|= |y|+|z|, but since B is not primitively generate* *d, there is no problem with QBp|x|being zero. Thus A = k[x]=(xp) since |x|is even if p > 2. As in case three, the exact sequence from theorem 2.6 and lemma 2.3 imply the* * elements of H 3(A; B) have representatives (oA; OEB ) with OEB trivial and oA determined by* * oA(y; z) - oA(z; y) = ax. Elements giving C noncommutative have a nonzero, and all will be non-primit* *ively generated. Scaling x allows us to assume a = 1. As before, we have x = x 1, y = 1 y, and z= 1 z as possible algebra genera* *tors, but the commutators are now [x; y] = [x; z] = 0 and [y; z] = x. Thus {xiyjzk} spans C. * *The coproduct calculation shows that x and yare primitive and (z) = X __1_yr ys; r+s=pr ! s ! r;s>0 which in turn implies that x and y have height p and that zp is in the span of * *{xiyjzk} for 0 i; j; k < p. Degree considerations and the fact that y commutes with zp show th* *at zp = axp-1y. To determine a, compare terms of the form - y in the coproduct of either side.* * A long but elementary computation with commutators and binomial coefficients gives a = 1. * *Thus we have the Hopf algebra of type (b). Note that the Hopf algebra of type (b) is essentially the sub-Hopf algebra of* * the Steenrod algebra generated by y= P 1and z= P p. Fields which are not perfect. The assumption that k be perfect in the statement of theorems 3.2 and 3.3 is * *necessary in order to use Borel's theorem to classify the possible algebra structures on these Hop* *f algebras. The theory of section two can also be used for this purpose. When this is done we see that* * Borel's theorem gives a complete list for total rank two, even if k is not perfect. Thus theore* *m 3.2 is true for general k. For total rank three the theory yields one additional algebra structure whe* *n k is not perfect. However, there is only one possible coproduct for this algebra structure, so th* *eorem 3.3 is true for general k if one more type of Hopf algebra is added to the list. That is the pr* *imitively generated Hopf algebra given by 2 k[x; y]=(yp ; xp - lyp); where l is nonzero in k. If k is perfect, the substitution x0= y - ax with l = * *ap shows that this is of type (g-1). For more general fields, some of these Hopf algebras may coincide f* *or certain values of l. Finally, theorem 3.9 remains true when k is not perfect since the assumption * *is only made in order to cite theorem 3.2. References. [1]J. W. Milnor and J. C. Moore, On the structure of Hopf algebras, Ann. of Mat* *h. (2) 81 (1965) 211-264. [2]W. M. Singer, Extension theory for connected Hopf algebras, J. Algebra 21 (1* *972) 1-16. [3]C. W. Wilkerson, The cohomology algebras of finite dimensional Hopf algebras* *, Trans. Amer. Math. Soc. 264 (1981) 137-150. Institut Mittag-Leffler Aurav"agen 17, S-182 62 Djursholm, Sweden Current address: Pennsylvania State University University Park, PA 16802