Showing papers on "Hopf algebra published in 1980"

The hopf algebra of linearly recursive sequences

Abstract: We explain how the space of linearly recursive sequences over a field can be considered as a Hopf algebra. The algebra structure is that of divided-power sequences, so we concentrate on the perhaps lesser-known coalgebra (diagonalization) structure. Such a sequence satisfies a minimal recursive relation, whose solution space is the subcoalgebra generated by the sequence. We discuss possible bases for the solution space from the point of view of diagonalization. In particular, we give an algorithm for diagonalizing a sequence in terms of the basis of the coalgebra it generates formed by its images under the difference-operator shift. The computation involves inverting the Hankel matrix of the sequence. We stress the classical connection (say over the real or complex numbers) with formal power series and the theory of linear homogeneous ordinary differential equations. It is hoped that this exposition will encourage the use of Hopf algebraic ideas in the study of certain combinatorial areas of mathematics.

On an analog of Lagrange’s theorem for commutative Hopf algebras

Abstract: We show that the Hopf algebra A over any field k representing the affine group scheme SL(2, ) is not a free B-module for some sub-Hopf algebra B of A. In particular k can be algebraically closed, or of characteristic 0 in which case A is also cosemisimple. Introduction. Let G be a finite group and suppose that H is a subgroup of G. If k is a field, then Lagrange's theorem can be formulated in terms of the group algebra k[G], namely k[G] is a free (left) k[H]-module. An obvious Hopf algebra analog would be the following: If A is a Hopf algebra over a field k and B is a sub-Hopf algebra of A, then A is a free (left) 5-module. This is not generally true; there exist commutative cocommutative counterexamples over certain nonalgebraically closed fields [1]. By the theorem of [3], the field of definition of a cocommutative counterexample can never be algebraically closed. In this paper we show that the Hopf algebra A over a field k representing the affine group scheme SL(2, ) is a finitely generated 5-module which is not free for some sub-Hopf algebra B of A. In particular, k can be algebraically closed. In characteristic 0 observe that A is cosemisimple [7]. Our proof is very elementary and reduces to simple computations in the polynomial ring k[X]. It should be noted that a commutative Hopf algebra A over a field k is always a faithfully flat and projective A-module for any sub-Hopf algebra B [5, Theorem 3.1] and [6, Corollary 1], and if B is finite-dimensional then A is a free A-module [3, Theorem 1]. For further results on the analog of Lagrange's theorem the reader is referred to [2], [3], [5], [6]. The main result. Let k be a field and n > 1 a fixed integer. Let C be the vector space with basis of symbols Xy, 1 < i,j < n. Then C is a coalgebra where AXy = 2* A",* 0 Xkj and e(Xy) = 8y for all i,j. The coalgebra structure on C extends uniquely to a bialgebra structure on the symmetric algebra S(C) of C. It is well known that the determinant form d = 2,asgn(o)XXaW... -X^^ satisfies Ad = d ® d, and the quotient A = S(C)/(d — 1) has a Hopf algebra structure. It is easily seen that A represents SL(n, ). For the remainder of this paper we let n = 2. Received by the editors June 9, 1979. AMS (MOS) subject classifications (1970). Primary 16A24.

A lower central series for split hopf algebras with involution

Abstract: A lower central series is defined for split Hopf algebras with involution over a field k. Various structure theorems for coalgebras and Hopf algebras are established. Introduction. In (6), Moore and Smith define a lower central series for homology Hopf algebras, i.e. connected Hopf algebras with commutative comultiplication. In essence their definition is a categorical one, the point being that the category of homology Hopf algebras possesses enough structure to apply the categorical construction. It is the object of this paper to show that a lower central series can likewise be defined for objects of %/k, the category of split Hopf algebras with involution over a field k. In order to accomplish this task it is necessary to expend much effort establishing the basic properties of %/k. §1 lays the foundation for studying split Hopf algebras with involution by developing important results concerning coal- gebras and comodules. §§2 and 3 present structure theorems for objects in %/k. The proof of the fact that epimorphisms are normal in %/k occupies all of §4. Finally in §5 we are in a position to define the commutator sub-Hopf algebra of an object of %/k and to study descending series. It is assumed that the reader is familiar with the work of Milnor and Moore (5). Results and constructions from (5) are utilized freely throughout this paper. Also we use the terms Hopf algebra, coalgebra, comodule, etc., in the sense of Milnor and Moore, and we follow their notation whenever possible. Acknowledgements. This paper is essentially the author's undergraduate thesis which was submitted to Princeton University in May, 1976. Professor John C. Moore was the faculty advisor for this work. He suggested many of the problems considered here and offered both encouragement and mathematical insight generously. The author's personal indebtedness to Professor Moore is great; the extent of his mathematical indebtedness should be apparent. 1. Coalgebras and comodules. In this section some elementary properties of coalgebras and comodules are established. In the first part of the section, we assume that a commutative ring K has been chosen for the ground ring and tensor products are taken over K. ®fiL / K will be used to denote the category of graded AT-modules. In the later portion of the section the ground ring is usually a field k.

On bialgebras which are simple Hopf modules

Abstract: This paper gives a module characterization of commutative or cocommutative Hopf algebras over a field. 0. Introduction. Let A be a bialgebra over a field k. Then A has a natural left A-Hopf module structure, and if A is a Hopf algebra, an easy calculation with the antipode shows that A is a simple Hopf module. We show that a commutative or cocommutative bialgebra over a field k which is a simple left Hopf module is a Hopf algebra. From this result we derive a module-theoretic characterization of commutative or cocommutative bialgebras over a field k which are Hopf algebras; namely such a bialgebra is a Hopf algebra if and only if all left A-Hopf modules are free (or (0)). Generally a commutative or cocommutative bialgebra A over a field k has a unique maximal subcoalgebra A(,) which is a Hopf algebra. In both cases A(,) can be described in terms of grouplike elements-the basic results of this paper are derivatives of elementary observations concerning grouplikes in certain bialgebras. The author wishes to thank the referee for his comments and suggestions, and wishes to express appreciation to Rutgers University for its hospitality during the time of the revision of this paper. 1. Preliminaries. In this section we show that any bialgebra A over a field k has a unique subcoalgebra A(M) maximal among the subcoalgebras D such that the inclusion iD E Hom(D, A) has an inverse in the convolution algebra. We will see that A(,) is characterized by its simple subcoalgebras. For a coalgebra C over k recall that the wedge product U A V of subspaces U, V c C is defined by U A V = A/-'(U 0 C + C 0 V). The wedge product of subcoalgebras is a subcoalgebra. LEMMA 1. Let C be a coalgebra over a field k, and suppose E, D', D" C C are subcoalgebras, E simple. (a) If E C E D, where D runs over a family of subcoalgebras of C, then E C D for some D. (b) If E C D'A D" then E C D' or E C D". Received by the editors July 18, 1979 and, in revised form, January 28, 1980. AMS (MOS) subject classifications (1970). Primary 16A24; Secondary 16A48, 16A64.

On the Uniqueness of Dyer-Lashof Operations on the Bott Periodicity Spaces

Abstract: shows that each of these spaces is an infinite loop space. Dyer-Lashof operations Q (r>0) are defined on the mod p homology (p prime) of any infinite loop space, depending on its infinite loop structure. If p=2, they are natural homology operations of degree r such that Q(x^)=x%9 G^J—O if r