Showing papers on "Hopf algebra published in 1972"

A correspondence between Hopf ideals and sub-Hopf algebras

Abstract: In this paper we show a bijective correspondence between normal Hopf ideals and sub-Hopf algebras of a commutative Hopf algebra over a field k. This gives a purely algebraic proof of the fundamental theorem [2, III, §3, no7] of the theory of affine k-groups.

Constructing sequences of divided powers. II

Abstract: In my Sequences of divided powers in irreducible, cocommutative Hopf algebras, I demonstrated the existence of extensions of sequences of divided powers over arbitrary fields, if certain coheight conditions are met. Here, I show that if the characteristic of the field does not divide n, every sequence of divided powers of length n 1, in a cocommutative Hopf algebra, has an extension that can be written as a polynomial in the previous terms. (An algorithm for finding these polynomials is given, together with a list of some of them.) Furthermore, I show that if one uses this method successively for constructing a sequence of divided powers over a primitive, the only obstructions will occur at powers of the characteristic of the field. Some of the basic definitions of this paper are the following: (1) If H is a Hopf algebra and 0 $ g E H, then g is a grouplike if Ag = g 0 g. (2) If h c H, then h is a primitive if Ah = h 0 1 + 1 ? h. (3) A Hopf algebra will be called irreducible if every nontrivial subcoalgebra contains a fixed, nontrivial subcoalgebra, i.e., if H is irreducible, the identity is the unique grouplike. (4) An irreducible Hopf algebra will be called graded, if there exists a set of subspaces {Hj}j? O such that (a) H= G`0H,; (b) Ho= 1 k; (c) AHiC E=o Hj? Hi-; (d) HiHjC Hi+j (5) A sequence of divided powers Ox = 1, 1x, 2x, , x is a set of elements in a cocommutative, irreducible Hopf algebra such that Altx = 0 o t-iX, for all 0 < t < n. Received by the editors December 8, 1970. AMS 1970 subject classiflcations. Primary 16A24.

Sequences of divided powers in irreducible, cocommutative Hopf algebras

Abstract: In Hopf algebras with one grouplike element, M. E. Sweedler showed that over perfect fields, sequences of divided powers in cocommutative, irreducible Hopf algebras can be extended if certain \"coheight\" conditions are met. Here, we show that with a suitable generalization of \"coheight\", Sweedler's theorem is true over nonperfect fields. (We also point out, that in one case Sweedler's theorem was false, and additional conditions must be assumed.) In the same paper, Sweedler gave a structure theorem for irreducible, cocommutative Hopf algebras over perfect fields. We generalize this theorem in both the perfect and nonperfect cases. Specifically, in the nonperfect case, while a cocommutative, irreducible Hopf algebra does not, in general, satisfy the structure theorem, the sub-Hopf algebra, generated by all sequences of divided powers, does. Some additional properties of this sub-Hopf algebra are also given, including a universal property. Some of the important definitions used in this paper are the following: (1) If H is a Hopf algebra and Q^g e H, then g is a grouplike if Ag=g <8> g(2) If he H, then h is a primitive if Ah = h (g> 1 +1 <8> h. Note that the primitives of H form a vector space. We will call it P(H). (3) A Hopf algebra H will be called irreducible if every nontrivial subcoalgebra contains a fixed, nontrivial subcoalgebra, i.e. if H is irreducible, it has a unique grouplike. (Note. For Hopf algebras, the terms \"coconnected\" and \"irreducible\" are synonymous.) (4) A sequence of divided powers °x= 1, 1x, 2x,..., lx is a set of elements in H such that Anx = 2\"=o '* ® n~'x, for all O^n^t. Notation. Throughout the rest of this paper k will be a field of char p > 0 and k the perfect closure of k. H will be an irreducible, cocommutative Hopf algebra over k, with augmentation e, diagonalization A, and identity 1. 8 will be the map [(e — I) (« — /)] ° A, where Iis the identity map on H and we identity k with k-l. All tensor products will be over k, unless otherwise noted. If K is any extension field of k, we view H (g> K as a Hopf algebra over K, by letting A„®K: H K®K H® KxH (8) H K equal A IK and by Received by the editors September 2, 1970. AMS 1970 subject classifications. Primary 16A24.

$H$-spaces which are co-$H$-spaces

Abstract: It is shown that a space, having the homotopy type of a CW complex of finite type, admitting both H-space and co-Hspace structures must have the homotopy type of a point or an n-sphere for n= 1, 3 or 7. Various people have considered spaces which admit both H-space and co-H-space structures. For example C. S. Hoo [5] showed that the set of homotopy classes of maps from a polyhedron to such a space forms a Moufang loop thereby extending results of C. W. Norman and R. O'Neill. On the other hand Curtis and Dugundji [2] showed that if a compact Lie group admits a H-cogroup structure then it has rank 1. Here we offer the following result. THEOREM. Let X have the homotopy type of a connected CW complex offinite type (of possibly infinite dimension). If X admits both an H-space and co-H-space structure then X has the homotopy type of S', S3, S7 or a point. REMARK. Adams and Walker [1] have exhibited a four-dimensional countably infinite CW complex T which is surprisingly both an EilenbergMac Lane space of type (Q, 3) and a Moore space of the same type. Since T is a suspension by its construction, we see that the hypothesis about finite type is essential to the theorem. To prove the theorem let us suppose that X is not contractible. This implies that J7*(X)0O (or else I7*(X)=O and the Whitehead Theorem would imply contractibility, since -r1(X)_H1(X) for H-spaces). Our first step is CLAIM 1. IfR is thefield Q or Zfi, p a prime, the Hopf algebra H*(X; R) is the exterior algebra on a single (necessarily primitive) generator of odd degree n. PROOF. We first observe that all cup-products of elements of positive degree in H*(X; R) vanish. To verify this folklore result note that by Received by the editors January 25, 1971. AMS 1969 subject classifications. Primary 5501. Key w,ords and phrases. H-space, co-H-space, CW complex. ? American Mathematical Society 1972

Invariant splittings in nonassociative algebras: A Hopf approach

Abstract: 0. Let (K A, e) be a coalgebra over the field k which is equipped with the structure of a unitary associative algebra by means of coalgebra morphisms m: V ®k V-+ Kand /i: k -• V. A = (K A, e, m, fi) is then a bialgebra and is a Hopf algebra if ideEndk(F) is invertible in the convolution structure [2, p. 71]. We will often confuse A with V. Recall that A* has a natural associative algebra structure relative to