Jordan Algebras of Gelfand–Kirillov Dimension One
Jordan Algebras of Gelfand–Kirillov Dimension One
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DOI：
10.1006/jabr.1996.0063
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发表时间：
1996-02
影响因子：
0.9
通讯作者：
C. Martínez;E. Zelmanov
中科院分区：
文献类型：
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作者：
C. Martínez;E. Zelmanov
自 LeedhamGreen 和 Newman 的第 6 篇论文以来，窄对象（例如 pro-p-群和 w x 有限宽度的有限余类的李代数）已得到深入研究。有限宽度的李代数为 Gelfand]Kirillov 维数 1。Gelfand]Kirillov 的结合代数 w x w x 维数 1 的结构已由 Small Ž w x 在系列论文 12 和 13 中阐明。等人。这又取决于伯格曼的工作，参见 1 。对于李代数来说，涉及环代数和嘉当型代数的分类仍然是一个悬而未决的问题。然而，在某些自然条件下，有限宽度的李代数具有有限的 Z 分级，因此通过 Tits]Kantor]Koecher 构造与 Jordan 系统相关。这些 Jordan 系统也是 Gelfand]Kirillov 维 1 的。在本文中，我们研究 Gelfand]Kirillov 维 1 的 Jordan 代数，并证明了与 Small 等人的定理类似的定理。和伯格曼。本文的结果将用于后续关于全能群和有限宽度李代数的论文中。 Ž 。令 A 为基础场 K 上的有限生成不一定关联代数。令 V 为生成 A 的有限维 K 向量空间，令 V n 表示 V 的 Ž 元素中长度为 F n 的所有乘积的线性跨度。A 的 Gelfand]Kirillov 维数表示为
Narrow objects such as pro-p-groups and Lie algebras of finite coclass of w x finite width have been studied intensively since the paper 6 of LeedhamGreen and Newman. Lie algebras of finite width are of Gelfand]Kirillov dimension 1. The structure of associative algebras of Gelfand]Kirillov w x w x dimension 1 has been clarified in the series of papers 12 and 13 by Small Ž w x. et al. which in turn depend on the work of Bergman see 1 . For Lie algebras such a classification involving loop algebras and algebras of Cartan type remains an open problem However, under some natural conditions, Lie algebras of finite width have finite Z-grading and thus are related to Jordan systems via Tits]Kantor]Koecher construction. These Jordan systems are also of Gelfand]Kirillov dimension 1. In this paper we study Jordan algebras of Gelfand]Kirillov dimension 1 and prove theorems analogous to those of Small et al. and Bergman. The results of this paper will be used in a subsequent paper on prounipotent groups and Lie algebras of finite width. Ž . Let A be a finitely generated not necessarily associative algebra over a ground field K. Let V be a finite dimensional K-vector space generating A and let V n denote the linear span of all products of length F n in Ž elements of V. The Gelfand]Kirillov dimension of A denoted as