David E. Radford

The structure of Hopf algebras with a projection

The smash product algebra and the smash coproduct coalgebra are well known in the context of Hopf algebras [ 1,5], and these notions can be viewed as being motivated by the semidirect product construction in the theory of groups and in the theory of afIine group schemes, respectively. Let

Finite dimensional cosemisimple Hopf algebras in characteristic 0 are semisimple

Finite dimensional cosemisimple Hopf algebras are a natural generalization of group algebras. Much of the representation theory of finite groups can be extended to such Hopf algebras. However, there are still many unanswered basic questions for such Hopf algebras. One such question is a conjecture mentioned by Kaplansky [I]: the square of the antipode of a finite dimensional cosemisimple Hopf algebra is the identity. In this paper we prove that the fourth power of the antipode is the identity in a finite dimensional cosemisimple Hopf algebra over a field of characteristic 0. In the course of proving this, we investigate two elements of the Hopf algebra which are roughly analogous to the sum of the group elements in a group algebra. We show that Kaplansky’s conjecture is equivalent to these two elements being equal. For a finite dimensional Hopf algebra A with antipode s over a field, let L(p)(q) =pq for p, q E A* denote the left module action of A* on itself. In this paper we study two elements of A defined by the trace function: the element ;i defined by p(A) = Tr(L(p) (s’)*) for all PE A*, and the element x defined by p(x) = Tr(L(p)) for all p E A*. We show that 2 is a left integral and use it to prove Theorem 3.3, the main result of this paper: a finite dimensional cosemisimple Hopf algebra over a field of characteristic 0 is also semisimple (hence its antipode has order 1, 2, or 4). We extend the first corollary to [3, Proposition 91 in Theorems 4.3 and 4.4 by finding necessary and sufficient conditions for s to be a non-zero left integral (in which case x = A), and also study how x relates to the structure of A. In particular, when (dim A) 1 # 0 the right ideal xA may be thought of as a measure of the extent to which s* # I. The elements ji and .K are two examples of elements of A associated with an endomorphism of A. In general, for f~ End(A) we define A, E A by p(A&=Tr(L(p)cf*) for all PEA *. Thus I? = AS2 and x = A,. In Section 2 267 0021-8693/88 33.00

Yetter-Drinfel'd categories associated to an arbitrary bialgebra

Abstract Various prebraided monoidal categories associated to a bialgebra over a commutative ring are studied and their relationships at various levels are examined. Generalizations of braided bialgebras are described and prebraided monoidal categories are associated with them. Three formally different braided monoidal categories can always be associated with any bialgebra over a commutative ring. These are not necessarily the same.

A natural ring basis for the shuffle algebra and an application to group schemes

In this paper we construct a (ring) basis for the shuffle algebra Sh(V) of a vector space V over a field K and show that it provides a computational method for the study of commutative pointed irreducible Hopf algebras (such Hopf algebras represent unipotent pro-afIine group schemes). Let {Q: x E X} be an indexed basis for V and S = (X) be the free semigroup generated by X. The a, E vzl @ ... @ van’s, where

Reflexivity and coalgebras of finite type

Tn this papes we study various finiteness conditions on a coalgebra C and the dual algebra C* of all linear functionals on C. C is a ru

INVARIANTS OF 3-MANIFOLDS DERIVED FROM FINITE DIMENSIONAL HOPF ALGEBRAS

~k~ coalgebra if every finite dirne~jon~ C*-module is rational; e~u~vale~tI~, ever>“continuous” linear functional on C* (i.e., every functional which vanishes on an ideal of C* having finite codimension) is induced by an element of C. IJsing topological considerations introduced in Section 1 together tx-ith a mx2commutative counterpart of the Hilbert “basis” theorem we show in Section 3 that a coalgebra C is reflexive if the algebra C* contains a dense subalgebra which is finitely generated. In particular w-e show that the cofree coaIgebra I(r’)% the shuffle coalgebra Sh(F) and the ~‘Bir~hoff-~~itt~~ coalgebra of divided powers B(V), are all reAexive coalgebras II-hen F is a finite dimensional vector space. Since any connected coalgebra C may be imbedded in the shufXe coalgebra Sh( F) (where si is the space of primitive elements of C), it then follows that a connected coalgebra C is reflexive if and on& if the space of primitive elements of C is finite dimensional. This condition simpiy means that C is of “‘finite type” in the foIlowing sense. Let C, be the sum of the simple subcoalgebras of a coalgebra C and define indu~t~~e~~an increasing coalgebm filtration bq’

On the antipode of a quasitriangular Hopf algebra

This paper studies invariants of 3-manifolds derived from certain finite dimensional Hopf algebras via regular isotopy invariants of unoriented links in the blackboard framing. The invariants are based on right integrals for these Hopf algebras. It is shown that the resulting class of invariants is definitely distinct from the class of Witten-Reshetikhin-Turaev invariants. The invariant associated with the quantum double of a finite group G is treated in this context, and is shown to count the number of homomorphisms of the fundamental group of the 3-manifold to the given finite group G.

Finiteness conditions for a hopf algebra with a nonzero integral

Let (A, R) be a quasitriangular Hopf algebra with antipode s in the category of vector spaces over a field k. In this paper we show that s2 is inner, which means s is bijective, and when A is finite-dimensional there is a grouplike element h E A such that s4(a) = h&r-” for all a E A. Suppose that A is any Hopf algebra with antipode s over the field k. Proposition 1 of Section 1 gives a sufficient condition for 3’ to be inner which we apply to quasitriangular Hopf algebras in Section 2. Its statement and proof are based on calculations found in [2, pp. 66671. Generally s2 need not be inner. Suppose that A is finite-dimensional. If s2 were inner, then all ideals of A would be invariant under s2, and hence all subcoalgebras of the dual Hopf algebra A* would be invariant under S*, where S = s* is the antipode of A*. There are examples [5] over algebraically closed lields in which the latter is not the case. Section 2 begins with a discussion of properties of quasit~angular Hopf algebras (A, R) over the held k used in this paper. Write R=CR(‘)C@R(~)EA@A and set u=CS(R’*‘)R(~). The main result of Section 2 is that u is invertible and s2(a) = uau-’ for all a E A. This result was established in [Z, Exercise 7.3.61 under the hypothesis that s is bijective. If A is any finite-dimensional Hopf algebra with antipode s over the field k, there are distinguished grouplike elements g E A and tl E A * which relate left and right integrals and which together describe s4. In Section 3 we consider s4 when (A, R) is a unite-dimensional quasit~angular Hopf algebra. Let h = IIU, where u =C s(R’*‘) R(l) is defined as above and u= s(u) -I. By relating h to g and ~1, we are able to show that h is a grouplike element and that s4(a) = hah-’ for all a E A. When A is unimodular we prove that h =g. Thus the product vu does not depend on R in the unimodular case. We assume that the reader has some familiarity with the elementary aspects of the theory of Hopf algebras. A recommended reference is [6].

Pointed Hopf algebras are free over Hopf subalgebras

In [l] Heyneman and the present author considered various finiteness conditions for a coalgebra and its linear dual algebra. The purpose of this paper is to specialize the ideas of [I] and [3, 41 to Hopf algebras with a nonzero left integral, and to study the antipode and one-dimensional ideals of the dual algebra. Suppose A is a Hopf algebra which has a nonzero left integral. Then if D CA is a finite-dimensional subcoalgebra, D(=) is finite-dimensional. In particular, A is locally finite. If the coradical A,, is a reflexive coalgebra, then A is right strongly reflexive (cofinite right ideals of A* are finitely generated). By the discussion of [ 1, Sect. 3.71 one can expect many Hopf algebras with a nonzero left integral over an infinite field to be reflexive-in particular, for many Hopf algebras with a nonzero left integral over an infinite field all finitedimensional right A*-modules are rational. Let Jr C A* (resp., ST) d enote the space of left (resp., right) integrals for A. If t: A + A is any injective bialgebra map then t*(Jr) = Jr. In particular s*(jJ = Jr where s is the antipode of A. Using this observation we prove that s is bijective if sL f (0). We d iscuss the connection between the one dimensional ideals of A* and the grouplikes of A. The conclusions we draw generalize some of the results of [S]. Let J-C A be an injective hull of k . 1 (as a left A-comodule). If s1 # (0) then we show JA, = A. If A,, is a Hopf subalgebra in addition then A = A, for some n (thus Rad A* is nilpotent). This paper could easily be formulated in terms of the theory of injectives in the category of right A-comodules, but we choose to use basic properties of idempotents for the sake of completeness.

On a coradical of a finite-dimensional Hopf algebra

Abstract It is well known that a commutative or cocommutative Hopf algebra is faithfully flat over any Hopf subalgebra. Examples of commutative cocommutative cosemisimple Hopf algebras have been found which are not free as modules over certain Hopf subalgebras. In this paper various sufficient conditions are given for a Hopf algebra to be free over a Hopf subalgebra. It is shown that a pointed Hopf algebra is free over any Hopf subalgebra (as a left or right module).