两个$3 \times 3$矩阵的非交换迹代数的定义关系
Defining Relations of Noncommutative Trace Algebra of Two $3 \times 3$ Matrices
摘要：
非交换（或混合）迹代数$T_{nd}$由$d$类属$n\times n$矩阵和由类属矩阵乘积的所有迹生成的代数$C_{nd}$生成，$n,d\geq 2$。已知在特征为0的域上，该代数是中心$C_{nd}$的多项式子代数$S$上的一个自由模。对于$n=3$和$d=2$，我们已经明确地找到了这样一个子代数$S$和一组自由生成元的$S$ -模$T_{32}$。给出了$T_{32}$作为代数的一组定义关系和相应理想的Groebner基。证明是基于简单的计算机计算与标准功能的枫树，明确介绍$C_{32}$的发电机和关系，和方法的一般线性群的表示理论。
英文摘要：
The noncommutative (or mixed) trace algebra $T_{nd}$ is generated by $d$ generic $n\times n$ matrices and by the algebra $C_{nd}$ generated by all traces of products of generic matrices, $n,d\geq 2$. It is known that over a field of characteristic 0 this algebra is a finitely generated free module over a polynomial subalgebra $S$ of the center $C_{nd}$. For $n=3$ and $d=2$ we have found explicitly such a subalgebra $S$ and a set of free generators of the $S$-module $T_{32}$. We give also a set of defining relations of $T_{32}$ as an algebra and a Groebner basis of the corresponding ideal. The proofs are based on easy computer calculations with standard functions of Maple, the explicit presentation of $C_{32}$ in terms of generators and relations, and methods of representation theory of the general linear group.
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