A Cayley-Dickson process for a class of structurable algebras
A Cayley-Dickson process for a class of structurable algebras
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DOI：
10.1090/s0002-9947-1984-0735416-2
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发表时间：
1984
影响因子：
1.3
通讯作者：
B. Allison;J. Faulkner
中科院分区：
文献类型：
--
作者：
B. Allison;J. Faulkner
在本文中，我们研究所有简单可构造代数的类，其性质是斜埃尔米特元素空间的维数为1。这些具有对合性的代数是在李代数构造的研究中出现的。简化代数是 2 X 2 矩阵代数的同位素。我们研究了凯莱-迪克森过程，用于合理构造类中的一些代数，包括除法代数和非约化非除代数。该过程的一个重要特例赋予了 28 维 4 次中心简单 Jordan 代数 © 的两个副本的直和，具有对合代数的结构。在准备工作中，我们获得了一个程序，可以将任何此类乔丹代数 © 的痕量零元素的空间 ©0 赋予 27 维异常乔丹代数的结构。域 I 上的 E1 型分裂简单李代数的 56 维不可约模 911 具有对合代数结构，可用于构造 £7 和 £8 型分裂简单李代数 [2,5 和 11]。为了研究这些类型的非分裂李代数，对 91Lj 的 f 形式进行有理（在不涉及基场扩展的意义上）构造是有意义的，其中 I 表示 f 的代数闭包，而 911 ̂ 的 f 形式是在 f 上对合 (&,') 的代数，使得 (&,~)'f = 9Hf。 911 的底层向量空间是所有 2 X 2 矩阵的向量空间，其中标量对角线项和非对角线项选自 27 维分割异常中心简单 Jordan 代数。 911 上的乘法与从 Zorn 构造中获得的分裂八元数代数 0 与 f 上的乘法惊人地相似 [9，第 17 页]。 142]。因此，很自然地会问是否存在用于构造 f 型 911 ̂ 的 Cayley-Dickson 过程，该过程类似于用于构造 0j [15，p. 15] 的 f 型的经典 Cayley-Dickson 过程。 45]。我们将看到，通过将 28 维 4 次中心简单 Jordan 代数的两个副本的直和赋予对合代数的结构，可以构造 911 f 的 f 形式（参见例 6.12）。这种结构是凯利-迪克森过程的一个特例，也是本文的主题。在[1]中，研究了一类称为结构代数的对合代数。上一段提到的矩阵代数911是编辑1983年4月1日收到的一个例子。1980年数学学科分类。初级 17A30；次要 17B60、17C20。 1 研究部分由 NSERC 拨款支持。 2 研究部分由 NSF 拨款支持。 ©1984 美国数学会 0025-5726/84 每页 1.00 美元 + 0.25 美元 185 许可或版权限制可能适用于再分发；参见 https://www.ams.org/journal-terms-of-use 186 B.N. ALLISON 和 J. R. FAULKNER 简单结构化代数 (&,')，使得空间 §((£,")= (x E tP|x = -x} 具有维度 1。在 §§1-4 中，我们研究所有简单结构化代数 (&,') 的类，使得 §((J,") 具有维度 1。任何这样的代数 (&,~) 都具有四次形式 v，这种四次形式给出了 Freudenthal 三重系统的结构，从而使我们能够利用 Ferrar 在此类系统上的早期工作，以具有 4 次 Jordan 范数的 Jordan 代数为起点。第 5 节中研究了 4 次 Jordan 范数的一个例子。作为第 5 节工作的结果，我们得到了 4 次可分离 Jordan 代数中的迹 0 元素的空间具有 3 次可分离 Jordan 代数的结构。特别是，从 28 维中心简单 4 次（特殊）Jordan 代数开始，我们在第 6 节中描述了产生简单结构化的 Cayley-Dickson 过程。代数 (&, ~) 使得 dipt S(6E, ~) = 1。在第 7 节中，我们表明，对于某些参数的选择，该过程会产生除法代数。我们还在第 7 节中讨论了在特征零域上构造异常中心简单李代数的使用。最后，在第 8 节中，我们给出了给出简化代数的过程的必要和充分条件。在整篇论文中，我们假设 I 是特征域 = £2 或 3。I 上的所有向量空间和代数都被假设为 I 上的有限维。代数不一定被假设为结合律，并且（李代数除外）所有代数都被假设拥有由 1 表示的乘法恒等式。如果 & 是代数且 x, y, z E 6B，我们写为 [x, y, z] = (xy)z — x(yz) 具有对合的代数 (6?,") 是代数 t? 以及周期二的反同构 x -» x。如果 (&, ') 是具有对合的代数，我们用 %(&,-)= {xE&\x = x} 和 §(#,") = {x £ &\x 表示埃尔米特和斜埃尔米特元素的空间= -x]，在这种情况下 & = %(&, ~) © S(ffi,")。最后，假设 T 是 t 上的向量空间，m 是一个非负整数 dbFi 线性。作者希望对审稿人的一些有用建议表示感谢。许可或版权限制可能适用于再分发；请参阅 https://www.ams.org/journal-terms-of-use A CAYLEY-DICKSON PROCESS 187 1。我们在本节中收集了有关可构造代数的事实，这些事实将在后面的部分中使用。如果对于 x, y, z, w E 6£，则 f 上的对合 ((£,") 代数称为可结构化代数，其中 V E Endt(6£) 定义为 Vxyz = {x, y, z) = (xy)z + (zy)x (zx)y。我们假设对于本节的其余部分，我们写 § = S(fcV) 和 % = %(&,')。定义 ^: & X 6t -» § by ^s>-,(jc, j) = x/ — jx。如果不存在歧义，我们写 \p — *Pt&,-y 请注意 \[/(s, 1) = 2s对于 s E §，因此 ^ 是满射。对于 x、z E &，通过 Ux :y — Vx z、Lxy = xy、Rxy = yx 和 Ux — Uxx 定义 L/._,、Lx、Rx 和 (/ E Endt(6E)。然后我们对 x、y E &、s E S 有以下恒等式（参见 [1, §1], [2, 引理） 2] 和 [3, §11]): (1.2) [s,x, y] = -[x,s, y] = [x, y, s], v'-V "jt,^ ~~ "y.sx A^x,>>).*' (1.5) s^(x, j)s =-i//(sx,5y), 和 0-6) LJJXtyL, = -UJXt,y。另外 0 = [KIf,, Fx>>,] = K{JWt}>J1 F^^, (by (1.1)) ，因此 (L?) ^=^ 对于 x, y E 6E。因此，如果 x E 6E 和 s E S，^{x,iJT,x},ix 'x,{jx,^,ijr) — * x,s{x,sx,x) \"J V'1^// ^{x,jx,x},5x + ^4,(x,{X,sx,x})Ls（通过（1-4））。
In this paper, we study the class of all simple structurable algebras with the property that the space of skew-hermitian elements has dimension 1. These algebras with involution have arisen in the study of Lie algebra constructions. The reduced algebras are isotopic to 2 X 2 matrix algebras. We study a Cayley-Dickson process for rationally constructing some algebras in the class including division algebras and nonreduced nondivision algebras. An important special case of the process endows the direct sum of two copies of a 28-dimensional degree 4 central simple Jordan algebra © with the structure of an algebra with involution. In preparatory work, we obtain a procedure for giving the space ©0 of trace zero elements of any such Jordan algebra © the structure of a 27-dimensional exceptional Jordan algebra. The 56-dimensional irreducible module 911 for the split simple Lie algebra of type E1 over a field I possesses the structure of an algebra with involution that can be used in the construction of the split simple Lie algebras of type £7 and £8 [2,5 and 11]. In order to study nonsplit Lie algebras of these types, it is of interest to have a rational (in the sense that base field extension is not involved) construction of f-forms of 91Lj, where I denotes the algebraic closure of f and a f-form of 911 ̂ is an algebra with involution (&,') over f such that (&,~)'f = 9Hf. The underlying vector space for 911 is the vector space of all 2 X 2 matrices with scalar diagonal entries and nondiagonal entries chosen from the 27-dimensional split exceptional central simple Jordan algebra. The multiplication on 911 is strikingly similar to the multiplication on the split octonion algebra 0 over f as obtained from the Zorn construction [9, p. 142]. Thus, it is natural to ask whether there is a Cayley-Dickson process for constructing f-forms 911 ̂ that is analogous to the classical Cayley-Dickson process for constructing f-forms of 0j [15, p. 45]. We will see that a f-form of 911 f can be constructed by endowing the direct sum of two copies of a 28-dimensional degree 4 central simple Jordan algebra with the structure of an algebra with involution (see Example 6.12). This construction is a special case of the Cayley-Dickson process that is the main subject of this paper. In [1], a class of algebras with involution called structurable algebras was studied. The matrix algebra 911 referred to in the previous paragraph is an example of a Received by the editors April 1, 1983. 1980 Mathematics Subject Classification. Primary 17A30; Secondary 17B60, 17C20. 1 Research supported in part by an NSERC Grant. 2 Research supported in part by an NSF Grant. ©1984 American Mathematical Society 0025-5726/84 $1.00 + $.25 per page 185 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 186 B.N. ALLISON AND J. R. FAULKNER simple structurable algebra (&,') such that space §((£,")= (x E tP|x = -x} has dimension 1. In §§1-4, we investigate the class of all simple structurable algebras (&,') such that §((J,") has dimension 1. Any such algebra (&,~) possesses a quartic form v that occurs naturally as the denominator in the inversion operation. This quartic form gives (&,') the structure of a Freudenthal triple system and thereby allows us to make use of earlier work of Ferrar on such systems. The Cayley-Dickson process studied in this paper has as its starting point a Jordan algebra possessing a Jordan norm of degree 4. These norms are studied in §5. An example of a Jordan norm of degree 4 is the generic norm on a degree 4 separable Jordan algebra. We obtain as a consequence of our work in §5 the fact that the space of trace 0 elements in a degree 4 separable Jordan algebra has the structure of a degree 3 separable Jordan algebra. In particular, starting with a 28-dimensional central simple degree 4 (special) Jordan algebra, a 27-dimensional exceptional central simple Jordan algebra is obtained. In §6 we describe the Cayley-Dickson process that produces simple structurable algebras (&, ~) such that dimt S(6E, ~) = 1. In §7, we show that for certain choices of the parameters the process produces division algebras. We also discuss in §7 the use of such algebras in the construction of exceptional central simple Lie algebras over a field of characteristic zero. Finally, in §8, we give necessary and sufficient conditions for the process to give reduced algebras. Before proceeding, we fix some conventions and notation. Throughout the paper, we assume that I is field of characteristic =£2 or 3. All vector spaces and algebras over I are assumed to be finite dimensional over I. Algebras are not necessarily assumed to be associative and (except for Lie algebras) all algebras are assumed to possess a multiplicative identity denoted by 1. If & is an algebra and x, y, z E 6B, we write [x, y, z] = (xy)z — x(yz). An algebra with involution (6?,") is an algebra t? together with an anti-isomorphism x -» x of period two. If (&, ') is an algebra with involution, we denote the spaces of hermitian and skew-hermitian elements by %(&,-)= {xE&\x = x} and §(#,") = {x £ &\x = -x], in which case & = %(&, ~) © S(ffi,"). Finally, suppose Tis a vector space over t and m is a nonnegative integer dbFis linear. The authors wish to express their gratitude to the referee for several helpful suggestions. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use A CAYLEY-DICKSON PROCESS 187 1. Structurable algebras. We collect in this section the facts about structurable algebras that will be used in later sections. An algebra with involution ((£,") over f is called a structurable algebra if for x, y, z, w E 6£, where V E Endt(6£) is defined by Vxyz = {x, y, z) = (xy)z + (zy)x (zx)y. We assume for the rest of the section that (6B,"") is a structurable algebra over t. We write § = S(fcV) and % = %(&,'). Define ^: & X 6t -» § by ^s>-,(jc, j) = x/ — jx. If no ambiguity exists, we write \p — *Pt&,-y Note that \[/(s, 1) = 2s for s E § and so ^ is a surjection. For x, z E &, define L/._,, Lx, Rx, and (/ E Endt(6E) by Ux :y — Vx z, Lxy = xy, Rxy = yx, and Ux — Uxx. Then we have the following identities for x, y E &, s E S (see [1, §1], [2, Lemma 2], and [3, §11]): (1.2) [s,x, y] = -[x,s, y] = [x, y, s], v'-V "jt,^ ~~ "y.sx A^x,>>).*' (1.5) s^(x, j)s =-i//(sx,5y), and 0-6) LJJXtyL, = -UJXt,y. Also 0 = [KIf,, Fx>>,] = K{JWt}>J1 F^^, (by (1.1)) and so (L?) ^=^ for x, y E 6E. Thus, if x E 6E and s E S, ^{x,iJT,x},ix 'x,{jx,^,ijr) — * x,s{x,sx,x) \"J V'1^// ^{x,jx,x},5x + ^4,(x,{X,sx,x})Ls (by (1-4)).