单位根情形下量子幂零代数的删除导子算法
A Deleting Derivations Algorithm for Quantum Nilpotent Algebras at Roots of Unity
AI总结：
本文将Cauchon的删除导子算法和典范嵌入扩展至允许单位根的量子代数，通过变量替换构造量子仿射空间，并给出完全素商代数的PI度数显式公式及最大维不可约表示的构造方法。
摘要：
本文将Cauchon的算法和典范嵌入扩展到一大类量子代数。它适用于满足适当假设的域上的迭代Ore扩张，这些假设涵盖了Cauchon原始设定的条件，同时允许单位根。该扩展算法通过在分式除环$\mathrm{Frac}(A)$内进行一系列变量替换，从原始量子代数$A$构造出量子仿射空间$A'$。典范嵌入将完全素理想$P\lhd A$映射为完全素理想$Q\lhd A'$，使得当$A$是PI代数时，${\rm PI}\text{-}{\rm deg}(A/P) = {\rm PI}\text{-}{\rm deg}(A'/Q)$。当量子参数是单位根时，我们可以给出完全素商代数的PI度数的显式公式。本文最后给出了一种方法：当$A$是PI代数时，给定$A'/Q$的一个合适的不可约表示，可以构造出$A/P$的一个最大维不可约表示。
英文摘要：
This paper extends an algorithm and canonical embedding by Cauchon to a large class of quantum algebras. It applies to iterated Ore extensions over a field satisfying some suitable assumptions which cover those of Cauchon's original setting but also allows for roots of unity. The extended algorithm constructs a quantum affine space $A'$ from the original quantum algebra $A$ via a series of change of variables within the division ring of fractions $\mathrm{Frac}(A)$. The canonical embedding takes a completely prime ideal $P\lhd A$ to a completely prime ideal $Q\lhd A'$ such that when $A$ is a PI algebra, ${\rm PI}\text{-}{\rm deg}(A/P) = {\rm PI}\text{-}{\rm deg}(A'/Q)$. When the quantum parameter is a root of unity we can state an explicit formula for the PI degree of completely prime quotient algebras. This paper ends with a method to construct a maximum dimensional irreducible representation of $A/P$ given a suitable irreducible representation of $A'/Q$ when $A$ is PI.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
