正则分级的素性性质
Primeness property for regular gradings
AI总结：
本文研究了正则分级的素性性质，证明了G-正则代数不满足素性，而Z_2-正则代数在普通意义上满足素性，且最小性对正则性无要求。
AI中文摘要：
设K为特征为0的代数闭域，G为有限阿贝尔群。对于一个G-分级的K-代数A，我们定义分级中心多项式的素性性质：对于任何在不相交变量集中的分级多项式f和g，如果fg是分级中心的，则f和g都是分级中心的。令A=⊕_{g∈G}A_g为其分解为同质分量。假设对于每一个n元组(g_1,…,g_n)∈G，存在a_i∈A_{g_i}使得a_1⋯a_n≠0，并且对于每个g,h∈G，存在一个标量β(g,h)∈K*使得a_ga_h=β(g,h)a_ha_g。则该分级是正则的，且若不存在不同的g,h∈G满足β(g,x)=β(h,x)对所有x∈G，则为最小的。我们证明G-正则代数，包括具有泡利分级的M_n(K)，不满足素性性质。对于阶数为2和3的矩阵，不存在非平凡的分级满足素性。最后，对于Z_2-正则代数，我们利用已知事实：最小的正则分级满足无限维格拉斯代数E的分级恒等式且包含E的一个副本，从而证明此类代数在普通意义上满足素性性质。作为结果，我们展示最小性对于分级的正则性并不必要。
英文摘要：
Let $K$ be an algebraically closed field of characteristic $0$ and $G$ a finite abelian group. For a $G$-graded $K$-algebra $A$, we define the primeness property for graded central polynomials: for any graded polynomials $f$ and $g$ in disjoint sets of variables, if $fg$ is graded central, then both $f$ and $g$ are graded central. Let $A=\bigoplus_{g\in G} A_g$ be its decomposition into homogeneous components. Assume that for every $n$-tuple $(g_1,\dots,g_n)$ in $G$, there exist $a_{i}\in A_{g_{i}}$ with $a_1\cdots a_n\neq 0$, and that for each $g$,$h\in G$ there exists a scalar $β(g,h)\in K^{\ast}$ such that $a_ga_h=β(g,h)a_ha_g$. Then the grading is regular, and minimal if no distinct $g$, $h\in G$ satisfy $β(g,x)=β(h,x)$ for all $x\in G$. We prove that $G$-graded regular algebras, including $M_n(K)$ with the Pauli grading, fail the primeness property. For matrices of orders $2$ and $3$, no nontrivial gradings satisfy primeness. Finally, for $\mathbb{Z}_2$-graded regular algebras, we use the known fact that minimal regular gradings satisfy the graded identities of the infinite-dimensional Grassmann algebra $E$ and contain a copy of $E$ to show that such algebras satisfy the primeness property in the ordinary sense. As a consequence, we show that minimality is not required for the regularity of the grading.
