Ore Extensions and Infinite Triangularization

TL;DR

This paper introduces Ore-solvable algebras, a broad class of operator algebras on infinite-dimensional spaces, and proves new triangularization results that extend classical theorems like Lie and Engel, with implications for algebra structure.

Contribution

It defines Ore-solvable algebras and establishes their triangularization properties, generalizing classical results and characterizing finite-dimensional triangularizable algebras.

Findings

Proves triangularization for Ore-solvable algebras.

Shows Ore-solvable algebras include many classical algebra types.

Characterizes finite-dimensional triangularizable algebras as Ore-solvable.

Abstract

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.