论文标题
$ \ mathbb {z} $ - 格拉斯曼代数的全部支持等级
$\mathbb{Z}$-gradings of full support on the Grassmann algebra
论文作者
论文摘要
让$ e $成为在特征零的场上$ f $上的无限尺寸格拉斯曼代数。在本文中，我们研究了$ \ mathbb {z} $的结构 - 全额支持的$ e $等级。使用基本数字理论的方法，我们描述了所谓的$ 2 $诱导的$ \ mathbb {z} $的$ \ mathbb {z} $ - 分级的多项式身份 - 在$ e $ e $ e $ e $ e $ e $的全部支持上。由于这一事实，我们提供了$ \ mathbb {z} $的示例 - $ e $上的分级，这些等级是pi-e quivalent的，但不是$ \ mathbb {z} $ isomorphic。这是带有无限支持的分级代数的第一个例子，该代数为pi均等，而不是等法代数为级代数。我们还介绍了$ e $的中央$ \ mathbb {z} $的概念，我们表明其$ \ mathbb {z} $ - 分级的多项式身份与$ \ mathbb {z} _ {2} $分级polynomial Identitions $ \ m mathbbbb {z $ e} $ - $ e} $ - } $ _____________________________________________2
Let $E$ be the infinite dimensional Grassmann algebra over a field $F$ of characteristic zero. In this paper we investigate the structures of $\mathbb{Z}$-gradings on $E$ of full support. Using methods of elementary number theory, we describe the $\mathbb{Z}$-graded polynomial identities for the so-called $2$-induced $\mathbb{Z}$-gradings on $E$ of full support. As a consequence of this fact we provide examples of $\mathbb{Z}$-gradings on $E$ which are PI-equivalent but not $\mathbb{Z}$-isomorphic. This is the first example of graded algebras with infinite support that are PI-equivalent and not isomorphic as graded algebras. We also present the notion of central $\mathbb{Z}$-gradings on $E$ and we show that its $\mathbb{Z}$-graded polynomial identities are closely related to the $\mathbb{Z}_{2}$-graded polynomial identities of $\mathbb{Z}_{2}$-gradings on $E$.
