Showing papers on "Hopf algebra published in 1975"

On a coradical of a finite-dimensional Hopf algebra

Abstract: Examples are constructed showing that the coradical of a finite-dimensional Hopf algebra over any algebraically closed field is not necessarily a subalgebra (hence the Jacobson radical is not a Hopf ideal in general) The square of the antipode may induce a permutation on the simple subcoalgebras of dimension > 1 of arbitrarily high order 0 Introduction In this paper we discuss the coradical (and thus nat- urally the Jacobson radical) of a finite-dimensional Hopf algebra over a field k Over any algebraically closed field finite-dimensional examples are constructed such that the coradical is not a sub-Hopf algebra (hence the Jacobson radical is not necessarily a Hopf ideal) In (2) it is shown that the square of the antipode maps each simple subcoalgebra into itself in the co- semisimple case In general this does not happen; there exist finite-dimen- sional Hopf algebras over any algebraically closed field k with antipode s of order 2b (b large enough) such that t = s induces a permutation on the simple subcoalgebras of dimension > 1 or order n The Hopf algebras in this paper arise surprisingly enough as dual Hopf algebras of pointed ones which have a relatively simple descrip- tion The basic ideas of the constructions are taken from (5) and (3) The moderate amount of notation and results concerning Hopf algebras we assume can be found in L4J, and the algebra we take for granted is adequately cov- ered in (l) All vector spaces, coalgebras, Hopf algebras, etc will be over a field k 1 Preliminaries The examples we construct in this paper will be dual Hopf algebras of finite-dimensional pointed Hopf algebras The basic building block is Taft's example of (5) as described below 11 Let k be a field containing a primitive Bth root of unity co, b > 1 The free algebra X = k\A, Z) on noncommuting indeterminants A and Z is a bialgebra where A: X —> X ® X is determined by

${\rm BP}\sb\ast({\rm BP})$ and typical formal groups

Abstract: It is shown that BP*(BP) represents the functor which assigns to a commutative Zζpγ algebra R the set of isomorphisms between typical formal groups over R. The structure maps of the Hopf algebra BP*(BP) all arise naturally from this point of view, and one can easily derive the formulas of Adams [2, Theorem 16.1] for them. 1) Research supported in part by an N.S.F. grant.

A commutative Noetherian Hopf algebra over a field is finitely generated

Abstract: Let k be an arbitrary field and H a commutative Hopf algebra over k. We give a short proof of the fact that H is Noetherian if and only if H is finitely generated as a k-algebra. In [4] M. Takeuchi has shown that a commutative (or cocommutative) Hopf algebra H is faithfully flat over any sub-Hopf algebra K. He then uses this result to give a simple proof of the following: (INJ) If H is a commutative Hopf algebra over a field k then the correspondence (sub-Hopf algebras) (Hopf ideals) K ~ K+H is injective. As a corollary to (INJ) Takeuchi shows [4, 3.11] that any sub-Hopf algebra of a finitely generated, commutative Hopf algebra over a field is finitely generated. The purpose of this note is to show that a commutative Noetherian Hopf algebra over a field is finitely generated, and that this fact follows from (INJ) with essentially the same proof as Takeuchi's corollary. For any Hopf algebra H over the field k, we let H+ = ker(c) where c is the counit of H, and we denote the antipode of H by S. A commutative Hopf algebra is said to be Noetherian (resp. finitely generated) if it is Noetherian (resp. finitely generated) as a k-algebra. For the basic facts about Hopf algebras see [1] or [3]. Clearly the desired result follows from the following Proposition. Let H be a commutative Hopf algebra over a field k. If H+ is finitely generated as an ideal then H is finitely generated. Proof. Let .f/,, be generators of the ideal H+. If C is the subcoalgebra of H generated by the fi, then C is finite dimensional [1, Received by the editors December 2, 1974 and, in revised form, April 1, 1975. AMS (MOS) subject classifications (1970). Primary 16A24; Secondary 13E05.

On coverings and hyperalgebras of affine algebraic groups

Abstract: We will continue to study relationships between etale group coverings and the hyperalgebra of a connected affine algebraic group. Let k be a field, and let G be a connected affine algebraic k-group. The Hopf algebra of representative functions on the hyperalgebra hy(G) will determine a pro-affine algebraic k-group G and a covering 7r: G -G. In the previous paper [2] we studied basic properties of the covering (G, ir) and explained how it plays an essential role in the theory of etale group coverings. Sullivan [5] gave another description of G in positive characteristic, and he provided a transparent proof of the fact that ir is a central extension in [7]. In this article, we will give more properties of (G, ir) and improve our previous results to characterize those affine algebraic k-groups which have universal group coverings. The properties of (G, ir) we want to prove are the following two:

Forms of certain hopf algebras

Abstract: Let Ψ be a field, G a finite group of automorphisms of Ψ, and Φ the fixed field of G. Let H be a Hopf algebra over Ψ. For g ∈ G we define a Hopf algebra Hg which has the same underlying vector space as H and modified operations and show that the tensor product (over Ψ) ⊗g ∈ G Hg has a Φ-form. As a consequence we see that if n>0 is an integer and Φ is a field of characteristic zero or p>0 with (n,p)=1, then there is a finite dimensional Hopf algebra over Φ with antipode of order 2n.

Hopf algebras with nonsemisimple antipode

Abstract: An example is given to show that the antipode of a finite dimensional Hlopf algebra over a field of prime characteristic p > 2 need not be semisimple. (For p = 2 examples were previously known.) The example is a pointed irreducible Hopf algebra H (with antipode S) of dimension p3 such that S S2. Radford [4] has recently shown that the antipode S of a finite dimensional Hopf algebra H over a field K has finite order. Consequently, if K is of characteristic zero then the antipode of H is semisimple. On the other hand, if K is of characteristic 2 then S is semisimple only if S = I. (For otherwise S has even order, say 2k, and so 0 = 52k _ 1= (5k _ 1)2.) In this note we show that S may fail to be semisimple for any characteristic p > 3. We do this by constructing a pointed irreducible Hopf algebra of dimension p3 over an arbitrary field of characteristic p > 3 in which the antipode has order 2p (and hence is not semisimple). A related problem is that of finding a bound for the order of S. In [7] the authors have shown that if H is pointed, if G(H) has exponent e and if Ho C Hi C ... C Hm = H is the coradical filtration then (S2, _ I), = 0. Thus if K has characteristic p and pn-1 1 remains an open question. Presented to the Society, January 17, 1974 under the title Some finite dimensional pointed Hopf algebras with nonsemisimple antipode; received by the editors October 25, 1973. AMS (MOS) subject classifications (1970). Primary 16A24. lThis research was partially supported by National Science Foundation grant GP-38518. 2This research was partially supported by National Science Foundation grant GP-33226. Copyright

On the homology of a tensor product of local rings

Abstract: For quotient rings and of a regular ring the connection between the Hopf algebras , and is investigated. In general, this connection is expressed by a spectral sequence. Criteria are obtained for Bibliography: 18 items.

Multiplicative operations in bp*bp

Abstract: where S is the Landweber-Novikov algebra. S has the added advantage of being a cocommutative Hopf algebra over Z. This paper does not remove this difficulty, but we will show that the monoid of multiplicative operations in BP*BP, (i.e. those operations which induce ring endomorphisms on BP*X for any space X)9 which we will denote by Γ{BP), has a submonoid analogous to the monoid of multiplicative operations in S.

The antipode of a finite-dimensional Hopf algebra over a field has finite order

Abstract: The purpose of this note is to indicate a proof of the fact that the order of the antipode of a finite-dimensional Hopf algebra over a field has finite order. It has been shown [2], [5] that the order of the antipode of an infinite-dimensional Hopf algebra is not necessarily finite; and the order of the antipode is finite in the finite-dimensional case if the Hopf algebra is unimodular [1] or pointed and the ground field has prime characteristic [6] . Using the bilinear form introduced and studied in [3] we prove that the order is finite for any finite-dimensional Hopf algebra over a field. The bilinear form, integrals, grouplike elements, one-dimensional ideals, and the antipode are all related in rather intriguing ways. The proof of the finite order theorem is based on an explicit formula describing the fourth power of the antipode, as suggested by Theorem 5.5 of [1]. Full details will appear elsewhere [4].