论文标题
阶的对称矩阵的约旦代数的2级多项式身份
2-graded polynomial identities for the Jordan algebra of the symmetric matrices of order two
论文作者
论文摘要
在$ k $上订单的对称矩阵的Jordan代数具有两个自然等级，$ \ Mathbb {Z} _2 $，订单2的环状组。我们描述了这两个等级的分级多项式身份时，当基本场是无限的，与2个均等的基础相同时，我们都在两个中表现出了两个。在其中一种情况下，我们执行一系列计算，以便将问题减少到与协会者打交道，而在另一种情况下，一种采用了不变理论的方法和结果。此外，我们将后者的分级扩展到$ \ mathbb {z} _2 $ - 在$ b_n $上加入$ b_n $，在尺寸$ n $的矢量空间中的对称双线性形式的jordan algebra（$ n = 1 $，2，\ dots，$ \ dots，$ \ infty $）。我们称此评分为\ textsl {stalarar}一个，因为它甚至仅由标量组成。作为副产品，我们获得了$ \ mathbb {z} _2 $ graded Identities $ b_n $的有限基础。实际上，最后一个结果描述了这对$（b_n，v_n）$的弱约旦多项式身份。
The Jordan algebra of the symmetric matrices of order two over a field $K$ has two natural gradings by $\mathbb{Z}_2$, the cyclic group of order 2. We describe the graded polynomial identities for these two gradings when the base field is infinite and of characteristic different from 2. We exhibit bases for these identities in each of the two cases. In one of the cases we perform a series of computations in order to reduce the problem to dealing with associators while in the other case one employs methods and results from Invariant theory. Moreover we extend the latter grading to a $\mathbb{Z}_2$-grading on $B_n$, the Jordan algebra of a symmetric bilinear form in a vector space of dimension $n$ ($n=1$, 2, \dots, $\infty$). We call this grading the \textsl{scalar} one since its even part consists only of the scalars. As a by-product we obtain finite bases of the $\mathbb{Z}_2$-graded identities for $B_n$. In fact the last result describes the weak Jordan polynomial identities for the pair $(B_n, V_n)$.
