论文标题
关于矩阵代数的等级数
On the Number of Gradings on Matrix Algebras
论文作者
论文摘要
我们确定有限基团在上限三角矩阵的代数上，确定了基本等级的同构类别的数量。结果，我们证明，对于有限的Abelian Group $ g $，数字$ e（g，m）的顺序是代数$ g $ - gradings of代数$ g $ - $ m_ {m_ {m}（\ mathbb {f}）$ m \ times m $ m $ m $ materize $ math $ $ b的$ m $ m $ m $ nize $ \ n $ \ n n egradized $ g $ gradings $ g $ g。一个有限的阿贝尔基团在代数闭合场上的上限三角矩阵代数上的等法等级类别数量的公式，还提供了对其特征的轻度限制。最后，如果$ g $是一个有限的阿贝利安集团，$ \ mathbb {f} $是一个代数封闭的字段，$ n（g，m）$是$ g $ - $ m_ {m_ {m}％{m}％（\ mathbb {f}）$ g $ gradings $ g $ gradings的数字（\ mathb {f}）$ g \ right \ vert！} m^{\ left \ vert g \ right \ vert -1} \ sim e（g，m）$。
We determine the number of isomorphism classes of elementary gradings by a finite group on an algebra of upper block-triangular matrices. As a consequence we prove that, for a finite abelian group $G$, the sequence of the numbers $E(G,m)$ of isomorphism classes of elementary $G$-gradings on the algebra $M_{m}(\mathbb{F})$ of $m\times m$ matrices with entries in a field $\mathbb{F}$ characterizes $G$. A formula for the number of isomorphism classes of gradings by a finite abelian group on an algebra of upper block-triangular matrices over an algebraically closed field, with mild restrictions on its characteristic, is also provided. Finally, if $G$ is a finite abelian group, $\mathbb{F}$ is an algebraically closed field and $N(G,m)$ is the number of isomorphism classes of $G$-gradings on $M_{m}% (\mathbb{F})$ we prove that $N(G,m)\sim\frac{1}{\left\vert G\right\vert !}m^{\left\vert G\right\vert -1}\sim E(G,m)$.
