On the relation of the monomial group with other algebraic structures

非交换代数几何I：单变量单项式方程

Non Commutative Algebraic Geometry I: Monomial Equations with a Single Variable

作为非交换代数几何系列首篇论文，该研究通过构建Makanin-Razborov图编码齐次方程组的齐次解，进而分析单变量单项式方程组的解集，为自由结合代数上方程组解集结构研究奠定基础。

Semisimple finite group algebra of a generalized strongly monomial group

The complete algebraic structure of semisimple finite group algebra of a generalized strongly monomial group is provided. This work extends the work of Broche and del R{\'ı}o on strongly monomial groups. The theory is complimented by an algorithm and is illustrated with an example.

Which group algebras cannot be made zero by imposing a single non-monomial relation?

For which groups $G$ is it true that for all fields $k$, every non-monomial element of the group algebra $k\,G$ generates a proper $2$-sided ideal? The only groups for which we know this are the torsion-free abelian groups. We would like to know whether it also holds for all free groups. It is shown that the above property fails for wide classes of groups: for every group $G$ that contains an element $g\neq 1$ whose image in $G/[g,G]$ has finite order (in particular, every group containing a $g\neq 1$ that itself has finite order, or that satisfies $g\in [g,G])$; and for every group containing…