论文标题
中央式代数的结构
Structure of centralizer algebras
论文作者
论文摘要
给定一个$ n \ times n $矩阵$ c $上的统一环$ r $，在整个$ n \ times n $矩阵环$ m_n（r）$中的$ c $的c $被称为主要centrallizer矩阵环，由$ s_n（c，r）$表示。我们调查其结构并证明：$（1）$如果$ c $是具有$ c $ - 免费点的可逆矩阵，或者如果$ r $没有零distiSor，而$ c $是jordan-similar矩阵，所有特征值是$ r $，则是$ m_n（r）的$ m_n（r），然后是$ m_n（r）的$ s_ in n $ s_ in nius nounius of Sensermension n N. kasch。 $（2）$如果$ r $是一个积分域，而$ c $是Jordan类似矩阵，则$ s_n（c，r）$是Graham和Lehrer的蜂窝$ R $ -R $ -Algebra。特别是，如果$ r $是代数关闭的字段，而$ c $是$ m_n（r）$中的任意矩阵，则$ s_n（c，r）$始终是蜂窝代数，而扩展$ s_n（c，r）\ subseteq m_n（r）$始终是可分离的frobenius Extresension。
Given an $n\times n$ matrix $c$ over a unitary ring $R$, the centralizer of $c$ in the full $n\times n$ matrix ring $M_n(R)$ is called a principal centralizer matrix ring, denoted by $S_n(c,R)$. We investigate its structure and prove: $(1)$ If $c$ is an invertible matrix with a $c$-free point, or if $R$ has no zero-divisors and $c$ is a Jordan-similar matrix with all eigenvalues in the center of $R$, then $M_n(R)$ is a separable Frobenius extension of $S_{n}(c,R)$ in the sense of Kasch. $(2)$ If $R$ is an integral domain and $c$ is a Jordan-similar matrix, then $S_n(c,R)$ is a cellular $R$-algebra in the sense of Graham and Lehrer. In particular, if $R$ is an algebraically closed field and $c$ is an arbitrary matrix in $M_n(R)$, then $S_n(c,R)$ is always a cellular algebra, and the extension $S_n(c,R)\subseteq M_n(R)$ is always a separable Frobenius extension.
