The Order of the Antipode of a Finite Dimensional Hopf Algebra is Finite
The Order of the Antipode of a Finite Dimensional Hopf Algebra is Finite
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DOI：
10.2307/2373888
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发表时间：
1976-01
影响因子：
1.7
通讯作者：
D. Radford
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文献类型：
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作者：
D. Radford
设 A 是域 k 上对映点 s 的有限维 Hopf 代数。对于 A 中的非零左积分 x，令 a E G (A*) = Alg(A, k) 对于所有 h EA 满足 xh = a(h)x，并令 a E G (A) 为 A* 的对应元素。那么s4(h)=a-'(a -ha-')a。由此证明了论文的主要结果：有限维Hopf代数的对映阶是有限的。如果 x 是 A 的任意左积分，则 s(x)= a -x。标量 a (a) 在 s2 的结构中起着重要作用。对于 A 的任何积分（左或右）x，我们证明 s2(x) = a(a)x。对于 0#X E k ，属于 X 和 X -'a (a) 的 s2 特征空间具有相同的维数。特别地，S2的特征值X1,*,Xr可以被描述为X7 1a (a),...,Xrla (a)。 S2 的不变因子具有依赖于 a (a) 的对称度。如果a(a)在地面场中没有平方根，则dim A、A的群元素的阶以及s2的最小多项式的次数都是偶数。如果dimA是奇数，则s2的特征值X满足X2=a(a)。对映体 s * 下 A* 不变量的一维理想与集合 { g E G (A): g-2= a) 一一对应。这是从描述 s* 对一维理想的作用的公式得出的。最后，当 n > 1 时，用 2n 阶对映体构造有限维单模示例。给出高度对称的 8 维示例 e，其对映体为 4 阶，并且具有 e 和 e * 是单模的性质。 0. 简介。众所周知（[4]，[7]），无限维 Hopf 代数的对映点的阶可能不是有限的。有限维 Hopf 代数的例子已被发现 [9]，当 n > 1 时，其对映点为 2n 阶。如果 A 是单模的，则有限维 Hopf 代数 A 的对映点已被证明具有有限阶 [3]，或者如果 A 是尖的并且地面场具有素数特征 [10]。使用[3]和[5]的技术我们展示了333版权？ 1976 年，约翰·霍普金斯大学出版社。手稿于 1973 年 10 月 11 日收到。美国数学杂志，卷。 98，第 2 期，第 333-355 页 此内容于 2016 年 10 月 5 日星期三 04:13:54 UTC 从 207.46.13.113 下载，所有使用均遵循 http://about.jstor.org/terms 334 DAVID E. RADFORD。域 k 上的任何有限维 Hopf 代数 A 都有有限阶的对映体。在第 1 节中，我们介绍了整篇论文中使用的非奇异双线性形式 /8 (, )。如果 s: A->A 是双射双代数映射，则对于某个 O#cE k，s t=S -1 （st i 是 s 相对于 /3( , ) 的转置）。如果 s 是满足 s t = ws -1 的任何线性自同构，则标量 X 在 s 对 A 的作用中起着核心作用。s 的不变因子具有依赖于 w 的对称度。对于 0# X E k，我们表明属于 X 和 X ` 的 s 的特征空间具有相同的维度。因此特征值 X1, ... 。 s 的 Xr 也是 x 'cX,7'o。表明dimA的均匀性和s的不变因子的程度与地场中co的平方根的存在有关。在第二节中，我们讨论对映体、一维理想和类群元素之间的联系。我们的分析基于[5]中给出的对映体的表征。像 a E G (A*) = Alg(A, k) 这样的独特群，对于所有 h E A（x 是 A 的非零左积分）满足 xh = a (h)x 及其对应群 a E G (A) 在对映体的研究中至关重要。例如 s(x) = a -x，其中 s 是 A 的对映体。a 位于 G (A) 的中心。 A*的一维理想集合与G(A)一一对应。我们推导出对映体对 A* 的一维理想的作用公式。利用这一点，我们证明了对映点下 A* 不变量的一维理想与 { g E G (A) g-2 = a) 一一对应。作为推论，当且仅当 G (A) 具有奇数阶时，A* 在对映点下具有唯一的一维理想不变量。最后我们证明 s2 的 co = a (a) （参见上面的第二段）。在第 3 节中，我们专门关注域 k 上的有限维 Hopf 代数 A 的对映点 s。我们的第一个重要结果是 s4 的公式：即 hEA 的 s4(h)=a'-(a -h -a-)a。由此证明了本文的主要定理：有限维Hopf代数的对映体阶是有限的。 （因此，对于域上有限生成的平坦 Hopf 代数，结果是正确的。）因此，在特征 0 中，s 的幂是半简单算子。本节的其余部分专门讨论标量 X = a (a) 对 Hopf 代数结构的影响。第 4 节专门介绍与本文和 [3] 的一些结果相关的示例。我们用 2n 阶对映体构造所有 n > 1 有限维幺模 Hopf 代数。对于任何给定的偶数整数 n，我们构造一个有限维 Hopf 代数 A，使得 G (A*) -Zn 和 a（第 2 节的区别元素）对应于 Zn 的任何预定元素。最后，我们找到一个高度对称的 8 维示例 e，其对映体为 4 阶，使得 e 和 e * 是单模的。此内容于 2016 年 10 月 5 日星期三 04:13:54 UTC 下载于 207.46.13.113 所有使用均遵循 http://about.jstor.org/terms 有限维 HOPF 代数对映体的顺序。 335 我们的符号和术语本质上是[1]、[5]和[7]的符号和术语。所有向量空间和代数都在域 k 上。 1. 双代数图。设 A 为有限维代数。 M= A* 是一个 A 双模，其中对于所有 mEM 和 a, beA，m a(b)=m(ab)=b m(a)。对于固定的 mEM，通过 8 (a, b) = m(ab) 在 A 上定义双线性形式。然后 /3 导出映射 /l, /3r:A-*M，其中 131(a)(b)=/3(a,b)=/3r(b)(a)。因此 /,r(a)=wam 且 f,3(a) = m a。下面的观察将在续集中重复使用。 1.1.设 A 为有限维代数，m EM=A*。那么以下等价： (a) M=A*m (b) /3 是非奇异的 (c) M=mA。如果 /3 是有限维向量空间 V 上的任何非奇异双线性形式，则每个线性 s: V-> V 对于所有 v,w E V 都有一个由 /3 (s (v), w)= ,8 (v,st(w)) 定义的（唯一）转置 st。现在假设 A 是有限维 Hopf 代数，0 # m E M 是左积分。根据 [8] 的 4.3 M = A m，因此相关的双线性形式 /3 (a, b) = m (ab) 是非奇异的。进一步假设 s A ->A 是双代数的双射映射。那么 s*(m) 就是左积分。由于对于某些 0#coEk，左积分的空间是 [8] s*(m)=wm 的 4.1 维。请注意，对于 a, b E A，/3 (s(a), b) = m (s(a)b) =m o s(as (b)) =com(as (b)) = /3 (a,cws'(b)) 这意味着：1.2。 s'= cis 在某些条件下，M = A m，并且对于某些 XC E k，s * (m) = wm，其中 s 是 A 的对映体（参见第 2 节推论 4）。对于本节的其余部分，V 将是具有非奇异双线性形式 /3 的有限维向量空间，并且 s: V-> V 将是满足 1.2 的线性自同构。如果 s 满足 1.2，则应注意到 s = (st)t；此内容于 2016 年 10 月 5 日星期三 04:13:54 UTC 从 207.46.13.113 下载，所有使用均须遵守 http://about.jstor.org/terms 336 DAVID E. RADFORD。
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.