论文标题
矩阵上的分级单位身份和几乎非分级等级
Graded monomial identities and almost non-degenerate gradings on matrices
论文作者
论文摘要
让$ f $是特征零的字段，$ g $为一个组，$ r $为代数$ m_n（f）$，带有$ g $ - 级别。 Bahturin和Drensky证明，如果$ r $是小学并且中性组件是交换性的，那么$ r $的分级身份从三种基本类型的身份和长度$ \ geq 2 $的单个类型的身份和单个单独的身份都受到函数$ f（n）$ n $的范围。在本文中，我们证明最好的上限是$ f（n）= n $，更普遍地证明，在$ n $上，$ m_n（f）$上的基本$ g $ g $的所有分级单一身份都来自$ n $。我们还研究不满足单一身份的等级，而是微不足道的标识，我们称之为几乎是非分级等级。对基质代数的非分级基本等级的描述缩短为具有交换性中性成分的基质代数上的非分级基本等级的描述。我们提供必要的条件，以便对$ r $的分级几乎是非分类的，并且我们将结果应用于单个身份，以描述所有几乎非分级$ \ mathbb {z} $ - $ n \ leq 5 $的$ m_n（f）$上的分级。
Let $F$ be a field of characteristic zero, $G$ be a group and $R$ be the algebra $M_n(F)$ with a $G$-grading. Bahturin and Drensky proved that if $R$ is an elementary and the neutral component is commutative then the graded identities of $R$ follow from three basic types of identities and monomial identities of length $\geq 2$ bounded by a function $f(n)$ of $n$. In this paper we prove the best upper bound is $f(n)=n$, more generally we prove that all the graded monomial identities of an elementary $G$-grading on $M_n(F)$ follow from those of degree at most $n$. We also study gradings which satisfy no monomial identities but the trivial ones, which we call almost non-degenerate gradings. The description of non-degenerate elementary gradings on matrix algebras is reduced to the description of non-degenerate elementary gradings on matrix algebras that have commutative neutral component. We provide necessary conditions so that the grading on $R$ is almost non-degenerate and we apply the results on monomial identities to describe all almost non-degenerate $\mathbb{Z}$-gradings on $M_n(F)$ for $n\leq 5$.
