Showing papers on "Hopf algebra published in 1976"

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.

Hopf Algebra Structure of mod 2 Cohomology of Simple Lie Groups

Abstract: The purpose of the present paper is to determine the Hopf algebra structure of the mod 2 cohomology JJ*(G; Z2) of each compact connected simple Lie group G. For classical type G, the Hopf algebra H* (G; Z2) is determined by Borel [6] and Baum-Browder [3], except the spinor groups Spin(n) and the semi-spinor groups ,5s (4m). For exceptional type G, it is determined by several authors [6], [8], [9], [15], except the case G = AdE7 = E7/Zz. In order to describe our results, we shall use the submodule TG* of /J*(G;Zp) which consists of the transgressive elements with respect to the fibering

The structure of Morava stabilizer algebras

Abstract: The purpose of this note is to prove some general theorems which will facilitate the computation of Ext*e.sp(BP , , v21BP./I.), where 1.=(p, vl, ..., v,_ 0 is the n-th invariant prime ideal in BP.. Specific calculations and applications to the Novikov spectral sequence will be exposed in [8] and 1-13]. This paper is a sequel to [4] in that we reprove some results of Morava ([10] and [11]) with more conventional algebraic topological methods. Our approach differs from those of Morava and Johnson-Wilson in that no use is made of any cohomology theories other than Brown-Peterson theory. Our results have the advantage of being more directly applicable to homotopy theoretic computations than Morava's were. Although none of his results are actually used here, this paper owes its existence to many inspiring and invaluable conversations with Jack Morava. I would also like to thank Haynes Miller, John Moore, Robert Morris, and Steve Wilson for their interest and help. In w 1 we use the change of rings theorem of [7] to show that computing the above mentioned Ext group is equivalent to computing the cohomology of a certain Hopf algebra S(n), which we call the Morava stabilizer algebra. We describe it explicitly using the results of [12]. In w we describe the relation of S(n) to a certain compact p-adic Lie group S. which Morava called the stabilizer group, as it was the isotropy group of a certain point in a scheme with a certain group action in [10]. This group has been studied to some extent by number theorists but we do not exploit this fact. Its basic cohomological properties were originally found by Morava and the author (very likely not for the first time) by application of the results of Lazard I-5]. The results of w 3, however, make no use of [5] or even of the existence of S., and w 3 is independent of w 2. We do however use this group theoretic interpretation to get a certain splitting (Theorem (2.12)) of S(n) when p does not divide n.

On Geroch’s counterexample to the algebraic Hopf conjecture

Abstract: In this paper we present a simplified version of Geroch's counterexample to the algebraic Hopf conjecture which requires only a knowledge of the formal symmetries of Riemannian curvature tensors. A number of questions in differential geometry have revolved around the algebraic Hopf conjecture: that in even dimensions, a curvature tensor with positive sectional curvatures has positive Gauss-Bonnet integrand. In dimension two the conjecture is trivial, and in dimension four has been proved by Milnor, see [1]. However, in dimension six and higher even dimensions, the conjecture has been shown to be false by Geroch [2]. Geroch's counterexample consists of an abstract algebraic construction, whose understanding requires careful thought. The purpose of this note is to provide an easily understood numerical example which is derived from Geroch's example. We define the components of a curvature tensor R on a. six dimensional vector space V with basis {e,e2e3e4e5e6} by ^1234 = °1256 = °3456 = ^1423 = °1625 = ^3645 = L ^1313 = ^1515 = ^2424 = ^2626 = ^3535 = ^4646 = ■>> ^1324 = ^1526 = °3546 = 2 with all components not derivable from these by the symmetry relations Kbcd = Bbaa Kbdc = Bcdab set to zero. It is trivial to check the Bianchi identities and thereby verify that R is indeed a curvature tensor. One may check that 2 a,Afl*Mtf**= 3[(«i*3 " «3*i + a2*4 " a4b2)2 itkh + (axb5 a5bx + a2b6 a6b2f + (a3b5 a5b3 + a4b6 a6b4)2] so all sectional curvatures are nonnegative. The Gauss-Bonnet integrand is fi = 2 £(')e(j)Ri(l)i(2)j(l)j(,2)RiO)i(4)jO)jWRi(5)i(6)j(5)j(6) U where i and j are permutations and e denotes their sign. Received by the editors July 7, 1975. AMS (MOS) subject classifications (1970). Primary 53A45, 53C20.

Eigenvalues of Hopf manifolds

Abstract: The eigenvalues of the Laplacians A and OI on the Hopf manifolds are described. Some isospectral results are also given.

A semidirect product decomposition for certain Hopf algebras over an algebraically closed field

Abstract: Let H be a finite dimensional Hopf algebra over an algebraically closed field. We show that if H is commutative and the coradical HO is a sub Hopf algebra, then the canonical inclusion HO -+ H has a Hopf algebra retract; or equivalently, if H is cocommutative and the Jacobson radical J(H) is a Hopf ideal, then the canonical projection H -+ H/J(H) has a Hopf algebra section. For a Hopf algebra H we denote the coradical (i.e. the sum of the simple subcoalgebras of H) by Ho, and the Jacobson radical by J(H). If 7: H -> K is a surjective (resp. injective) Hopf algebra map we say it splits if there exists a Hopf algebra map T: K -* H with 77 ? T = IK (resp. T ? 77 = IH). The purpose of this paper is to prove that if H is a finite dimensional Hopf algebra over an algebraically closed field we have the following: (A) If H is commutative and Ho is a sub Hopf algebra, then the canonical inclusion Ho -H splits as a map of Hopf algebras; or equivalently, (B) If H is cocommutative and J(H) is a Hopf ideal, then the canonical projection H -H/J(H) splits as a map of Hopf algebras. If follows from the results of [3] that the existence of a Hopf algebra splitting in (A) or (B) induces a semidirect product decomposition of the Hopf algebra H, and that such splittings are necessarily unique. For the standard facts about Hopf algebras see [1] or [7]; for splittings and exact sequences see [3]. It is easy to see that (A) and (B) are equivalent, for by finite dimensionality we have J(H*) = (H0)' and so Ho _ (H*/J(H*))*. Thus a splitting in one case induces a splitting in the other by transposing. We shall verify (B). We begin by establishing a special case of (B) which is valid over any field. If G is a group, let k[G] denote the group algebra of G over k. PROPOSITION 1. Let H = k[G] where G is a finite group and k is any field. If J(H) is a Hopf ideal of H then the canonical projection g: H -H/J(H) splits as a map of Hopf algebras. PROOF. If the characteristic of k is zero (or is relatively prime to the order Received by the editors January 20, 1976. AMS (MOS) subject classifications (1970). Primary 16A24; Secondary 13E10.

Hopf invariants for reduced products of spheres

Abstract: Let S,n be the mth reduced product complex of the even dimensional sphere Sn. Using 'cup'-products, James defined a Hopf invariant homomorphism Hm: 9mn_l(Sm-l ) _Z such that H2 is the classical Hopf invariant. Extending the result of Adams on H2 we determine the image of Hmn. Partial calculations were made by Hardie and Shar. 1. Hopf invariants and higher order Whitehead products. In [9] James proved, that the reduced product complex S' of the sphere S' is homotopy equivalent to the loop space OS"'+. The complex S' has a natural CW decomposition sn = Sn U e2n U . . .U emn U . . . the mn skeleton of which is denoted by Smn, Sn = sn. In this paper we will be concerned with the homomorphism Hmn Smn Smn-l I Z where n is even and m > 2. This homomorphism is a generalization due to James [10] of Steenrod's definition of the Hopf invariant [21]: Let a E Mn -I(Smn1); one can choose generators a1, am-, and x of dimension n, (m 1)n and mn respectively in the integral cohomology of the complex Sm-I Ua E mn. Then the Hopf invariant Hmn(a) is defined to be the integer for which a1 U am 1 = Hmn(a)x. Hn is the classical Hopf invariant [8]. Let the element [in]m E 7mn(Smn-1) be given by an attaching map of the cell emn in Sn. This element is an mth order Whitehead product [16], [6]. For example [ij]2 is the Whitehead product [in, in] of a generator in e SJn(Sn). It is well known that H2n([in, in]) = 2 and more generally Hmn([in]m) = m; see ?3. On the other hand we have H (Un) = 1 for the Hopf elements an E (Sn ), n = 2, 4, 8 [8] and im Hn = 2Z if n =# 2, 4, 8 by the celebrated theorem of Adams on Hopf invariants [2]. Moreover Toda showed in [23, p. 175] that for a prime number p, there exist ap E 7&2p "1(S2 1) (and a2 = (2 if p = 2) such that Hp (ap) = 1. We will prove that the elements ap and the Hopf elements a2, a4, a8 are the only elements of Hopf invariant one.