The Order of the Antipode of a Finite Dimensional Hopf Algebra is Finite

Abstract

Let A be a finite dimensional Hopf algebra over a field k with antipode s. For a nonzero left integral x in A let a E G (A*) = Alg(A, k) satisfy xh = a(h)x for all h EA, and let a E G (A) be the corresponding element for A*. Then s4(h)=a-'(a -ha-')a. From this we prove the main result of the paper: the order of the antipode of a finite dimensional Hopf algebra is finite. If x is any left integral of A then s(x)= a -x. The scalar a (a) plays a significant role in the structure of s2. For any integral (left or right) x of A we prove that s2(x) = a(a)x. For 0#X E k the eigenspaces of s2 belonging to X and X -'a (a) have the same dimension. In particular the eigenvalues X1,* , Xr of S2 can be described as X7 1a (a), .,Xrla (a). The invariant factors of S2 possess a degree of symmetry dependent on a (a). If a (a) has no square root in the ground field, then dim A, the order of the grouplike elements of A, and the degree of the minimal polynomial of s2 are all even. If dimA is odd there is an eigenvalue X of s2 satisfying X2 =a(a). The one dimensional ideals of A* invariant under the antipode s * are in one-one correspondence with the set { g E G (A): g-2= a). This follows from a formula describing the action of s* on a one dimensional ideal. Finally, finite dimensional unimodular examples are constructed with antipode of order 2n for n > 1. A highly symmetric 8 dimensional example e is given with antipode of order 4 and having the property that e and e * are unimodular. 0. Introduction. It is well known ([4], [7]) that the order of the antipode of an infinite dimensional Hopf algebra may not be finite. Examples of finite dimensional Hopf algebra have been found [9] which have antipode of order 2n for n > 1. The antipode of a finite dimensional Hopf algebra A has been shown to have finite order if A is unimodular [3], or if A is pointed and the ground field is of prime characteristic [10]. Using the techniques of [3] and [5] we show 333 Copyright ? 1976 by Johns Hopkins University Press. Manuscript received October 11, 1973. American Journal of Mathematics, Vol. 98, No. 2, pp. 333-355 This content downloaded from 207.46.13.113 on Wed, 05 Oct 2016 04:13:54 UTC All use subject to http://about.jstor.org/terms 334 DAVID E. RADFORD. that any finite dimensional Hopf algebra A over a field k has antipode of finite order. In Section 1 we introduce the nonsingular bilinear form /8 (, ) which is used throughout the paper. If s: A->A is a bijective bialgebra map then s t=S -1 for some O#cE k (st i the transpose ofs with respect to /3( , )). Ifs is any linear automorphism satisfying s t = ws -1, the scalar X plays a central role in the action of s on A. The invariant factors of s possess a degree of symmetry dependent on w. For 0# X E k we show that the eigenspaces of s belonging to X and X ` have the same dimension. Thus the eigenvalues X1, ... . Xr of s are also x 'cX,7'o. The eveness of dimA and the degree of the invariant factors of s is shown to be related to the existence of a square root of co in the ground field. In Section 2 we discuss the connection between the antipode, one dimensional ideals, and the grouplike elements. Our analysis rests on the characterization of the antipode given in [5]. The unique grouplike a E G (A*) = Alg(A, k) satisfying xh = a (h)x for all h E A (x a nonzero left integral of A) and its counterpart a E G (A) are of central importance in the study of the antipode. For example s(x) = a -x, where s is the antipode of A. a is in the center of G (A). The set of one dimensional ideals of A* is in one-one correspondence with G (A). We derive a formula for the action of the antipode on one dimensional ideals of A*. Using this we show that the one dimensional ideals of A* invariant under the antipode are in one-one correspondence with { g E G (A) g-2 = a). As a corollary, A* has a unique one dimensional ideal invariant under the antipode if and only if G (A) has odd order. Finally we show that co = a (a) for s2 (see second paragraph above). In Section 3 we focus exclusively on the antipode s of a finite dimensional Hopf algebra A over a field k. Our first important result is a formula for s4:namely s4(h)=a'-(a -h -a-)a for hEA. From this we prove the main theorem of the paper: the order of the antipode of a finite dimensional Hopf algebra is finite. (The result is thus true for finitely generated flat Hopf algebras over a domain.) As a consequence, in characteristic 0 the powers of s are semisimple operators. The remainder of the section is devoted to implications of the scalar X = a (a) to the structure of the Hopf algebra. Section 4 is devoted to examples which are relevant to some of the results of this paper and [3]. We construct for all n > 1 finite dimensional unimodular Hopf algebras with antipode of order 2n. For any given even integer n we construct a finite dimensional Hopf algebra A such that G (A*) -Zn and a (the distinguished element of Section 2) corresponds to any predetermined element of Zn. Finally we find a highly symmetric 8 dimensional example e with antipode of order 4 such that e and e * unimodular. This content downloaded from 207.46.13.113 on Wed, 05 Oct 2016 04:13:54 UTC All use subject to http://about.jstor.org/terms ORDER OF THE ANTIPODE OF A FINITE DIMENSIONAL HOPF ALGEBRA. 335 Our notation and terminology is essentially that of [1], [5], and [7]. All vector spaces and algebras are over a field k. 1. Bialgebra Maps. Let A be a finite dimensional algebra. M= A* is an A-bimodule where m a(b)=m(ab)=b m(a) for all mEM and a, beA. For a fixed mEM define a bilinear form on A by 8 (a, b) = m(ab). Then /3 induces maps /l, /3r:A-*M where 131(a)(b)=/3(a,b)=/3r(b)(a). Thus /,r(a)=wam and f,3(a) = m a. The following observation will be used repeatedly in the sequel. 1.1. Let A be a finite dimensional algebra and m EM=A*. Then the following are equivalent: (a) M=A*m (b) /3 is nonsingular (c) M=mA. If /3 is any nonsingular bilinear form on a finite dimensional vector space V then every linear s: V-> V has a (unique) transpose st defined by /3 (s (v), w)= ,8 (v,st(w)) for all v,w E V. Now suppose that A is a finite dimensional Hopf algebra and 0 # m E M is a left integral. By 4.3 of [8] M = A m, so the associated bilinear form /3 (a, b) = m (ab) is nonsingular. Assume further that s A ->A is a bijective map of bialgebras. Then s*(m) is a left integral. Since the space of left integrals is one dimensional by 4.1 of [8] s*(m)=wm for some 0#coEk. Notice that /3 (s(a), b) = m (s(a)b) =m o s(as (b)) =com(as (b)) = /3 (a,cws'(b)) for a, b E A which implies: 1.2. s'= cisUnder certain conditions M = A m, and s * (m) = wm for some XC E k, where s is the antipode of A (see Corollary 4 of Section 2). For the remainder of this section V will be a finite dimensional vector space with nonsingular bilinear form /3, and s: V-> V will be a linear automorphism satisfying 1.2. If s satisfies 1.2 one should notice that s = (st)t; for This content downloaded from 207.46.13.113 on Wed, 05 Oct 2016 04:13:54 UTC All use subject to http://about.jstor.org/terms 336 DAVID E. RADFORD.