论文标题
矿石扩展和无限三角化
Ore Extensions and Infinite Triangularization
论文作者
论文摘要
我们给出了无限三角化和严格的三角化结果，用于无限尺寸矢量空间上的操作员的代数。我们介绍了一类我们称为矿石可溶解的代数代数的代数：这些类似于迭代的矿石扩展，但不必自由地作为中间子线上的模块。矿石可溶解的代数包括许多示例，例如特定情况，例如多环基组的组代数或有限溶液组，包裹可解决的lie代数代数，量子平面和量子矩阵的代数。我们证明了该类别的三角形和严格的三角化结果，并展示了它们如何推广和扩展经典的同时同时三角形结果，例如Lie和Engel定理。我们表明，从某种意义上说，这些结果是最好的，表明任何有限的三角形代数都必须是这种类型的。我们还给出了严格的三角形与零和nilpotent代数之间的连接，并证明了通过递归“矿石”程序从构建块开始的代数的非常普遍的结果，该构建块是NIL，可交换或有限的尺寸代数。
We give infinite triangularization and strict triangularization results for algebras of operators on infinite dimensional vector spaces. We introduce a class of algebras we call Ore-solvable algebras: these are similar to iterated Ore extensions but need not be free as modules over the intermediate subrings. Ore-solvable algebras include many examples as particular cases, such as group algebras of polycyclic groups or finite solvable groups, enveloping algebras of solvable Lie algebras, quantum planes and quantum matrices. We prove both triangularization and strict triangularization results for this class, and show how they generalize and extend classical simultaneous triangularization results such as the Lie and Engel theorems. We show that these results are, in a sense, the best possible, by showing that any finite dimensional triangularizable algebra must be of this type. We also give connections between strict triangularization and nil and nilpotent algebras, and prove a very general result for algebras defined via a recursive "Ore" procedure starting from building blocks which are either nil, commutative or finite dimensional algebras.
