论文标题
具有退化迹线和痕量身份的矩阵代数
Matrix algebras with degenerate traces and trace identities
论文作者
论文摘要
在本文中，我们在多项式身份理论的框架中研究了具有退化痕迹的矩阵代数。第一部分专门研究$ n \ times n $对角矩阵的代数$ d_n $。我们证明，在堕落的痕迹的情况下，其所有痕量身份都取决于通勤定律和纯正的痕量身份。此外，我们将带有简并跟踪的$ d_ {n+1} $的跟踪身份与$ d_n $的跟踪与相应的跟踪相关联。这使我们能够确定$ d_3 $的跟踪T理想的发电机。 在第二部分中，我们研究了$ m_k（f）$的可交换子级别的$ c_k $ $ f + j $的$ c_k $，可以赋予所谓的奇怪痕迹：$ tr（a + j）=αaa +βJ$，用于任何$ a + j in c_k $ in c_k $ in c_k $，$ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a。这里$ j $是$ c_k $的根本。如果$β= 0 $这样的跟踪是退化的，我们研究代数$ c_k $所满足的痕量身份，每$ k \ geq 2 $。此外，我们证明这些代数会产生所谓的多项式生长的最小品种。 在本文的最后一部分中，专门研究多项式生长的品种，我们完全对同一作者的早期论文中引入了几乎多项式生长的代数种类的种类。
In this paper we study matrix algebras with a degenerate trace in the framework of the theory of polynomial identities. The first part is devoted to the study of the algebra $D_n$ of $n \times n$ diagonal matrices. We prove that, in case of a degenerate trace, all its trace identities follow by the commutativity law and by pure trace identities. Moreover we relate the trace identities of $D_{n+1}$ endowed with a degenerate trace, to those of $D_n$ with the corresponding trace. This allows us to determine the generators of the trace T-ideal of $D_3$. In the second part we study commutative subalgebras of $M_k(F)$, denoted by $C_k$ of the type $F + J$ that can be endowed with the so-called strange traces: $tr(a+j) = αa + βj$, for any $a+j \in C_k$, $α$, $β\in F$. Here $J$ is the radical of $C_k$. In case $β= 0$ such a trace is degenerate, and we study the trace identities satisfied by the algebra $C_k$, for every $k \geq 2$. Moreover we prove that these algebras generate the so-called minimal varieties of polynomial growth. In the last part of the paper, devoted to the study of varieties of polynomial growth, we completely classify the subvarieties of the varieties of algebras of almost polynomial growth introduced in an earlier paper of the same authors.
