Journal article
Ore extensions and infinite triangularization
Journal of algebra, Vol.572, pp.297-325
04/15/2021
DOI: 10.1016/j.jalgebra.2020.12.013
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.
Details
- Title: Subtitle
- Ore extensions and infinite triangularization
- Creators
- Jeremy R Edison - University of IowaMiodrag C Iovanov - Romanian AcademyAlex Sistko - Manhattan University
- Resource Type
- Journal article
- Publication Details
- Journal of algebra, Vol.572, pp.297-325
- DOI
- 10.1016/j.jalgebra.2020.12.013
- ISSN
- 0021-8693
- eISSN
- 1090-266X
- Publisher
- Elsevier Inc
- Grant note
- DOI: 10.13039/100000893, name: Simons Foundation
- Language
- English
- Date published
- 04/15/2021
- Academic Unit
- Mathematics
- Record Identifier
- 9984241051902771
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