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.

中文翻译：

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矿石扩展和无限三角化

我们给出了无限维向量空间上算子代数的无限三角化和严格三角化结果。我们引入了一类我们称为矿石可解代数的代数：它们类似于迭代矿石扩展，但不需要作为中间子环上的模块免费。矿石可解代数包括许多特殊情况的例子，如多环群或有限可解群的群代数、可解李代数的包络代数、量子平面和量子矩阵。我们证明了此类的三角化和严格三角化结果，并展示了它们如何推广和扩展经典的同时三角化结果，例如 Lie 和 Engel 定理。我们表明，这些结果在某种意义上是最好的，通过证明任何有限维可三角化的代数都必须是这种类型。我们还给出了严格三角化与零和幂零代数之间的联系，并证明了通过递归“Ore”过程定义的代数的非常普遍的结果，该过程从零、可交换或有限维代数的构建块开始。