Ore extensions and infinite triangularization

JOURNAL OF ALGEBRA

卷:572

Article

Edison, Jeremy R.^1,2  Iovanov, Miodrag C.^2,3  Sistko, Alex^2,4 

[1] Mt Mary Univ, Dept Math, Fidelis Hall, Milwaukee, WI 53222 USA

[2] Univ Iowa, Dept Math, MacLean Hall, Iowa City, IA 52242 USA

[3] Romanian Acad, Simion Stoilow Inst, Bucharest 010702, Romania

[4] Manhattan Coll, Dept Math, Res & Learning Ctr, Riverdale, NY 10471 USA

关键词: Triangularization;    Ore algebras;    Ore extensions;    Ore solvable algebras;    Locally-finite module;    Semiartinian module;    Strict triangularization;    Nil algebra;   

DOI  :  10.1016/j.jalgebra.2020.12.013

来源: Elsevier

【 摘 要 】

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. (C) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

Free