The quantum plane is the non-commutative polynomial algebra in variables x and y with $xy=qyx$. In this paper, we study the module variety of n-dimensional modules over the quantum plane, and provide an explicit description of its irreducible components and their dimensions. We also describe the irreducible components and their dimensions of the GIT quotient of the module variety with respect to the conjugation action of ${\mathrm{GL}}_n$.

中文翻译：

---

量子平面上的各种模块

量子平面是变量x和y的非交换多项式代数，其中$Xÿ=qÿX$。在本文中，我们研究了量子平面上n维模块的模块多样性，并对其不可约成分及其维数进行了明确描述。我们还描述了模块变体的GIT商的不可约成分及其维数相对于共轭作用的大小。${\mathrm{GL}}_{ñ}$。