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Infinite-dimensional triangularizable algebras

Infinite-dimensional triangularizable algebras AbstractLet Endk⁢(V){\mathrm{End}_{k}(V)}denote the ring of all linear transformations of an arbitrary k-vector space V over a field k. We define X⊆Endk⁢(V){X\subseteq\mathrm{End}_{k}(V)}to be triangularizable if V has a well-ordered basis such that X sends each vector in that basis to the subspace spanned by basis vectors no greater than it. We then show that an arbitrary subset of Endk⁢(V){\mathrm{End}_{k}(V)}is strictly triangularizable (defined in the obvious way) if and only if it is topologically nilpotent. This generalizes the theorem of Levitzki that every nilpotent semigroup of matrices is triangularizable. We also give a description of the triangularizable subalgebras of Endk⁢(V){\mathrm{End}_{k}(V)}, which generalizes a theorem of McCoy classifying triangularizable algebras of matrices over algebraically closed fields. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

Infinite-dimensional triangularizable algebras

Forum Mathematicum , Volume 31 (1): 15 – Jan 1, 2019

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Publisher
de Gruyter
Copyright
© 2018 Walter de Gruyter GmbH, Berlin/Boston
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/forum-2018-0109
Publisher site
See Article on Publisher Site

Abstract

AbstractLet Endk⁢(V){\mathrm{End}_{k}(V)}denote the ring of all linear transformations of an arbitrary k-vector space V over a field k. We define X⊆Endk⁢(V){X\subseteq\mathrm{End}_{k}(V)}to be triangularizable if V has a well-ordered basis such that X sends each vector in that basis to the subspace spanned by basis vectors no greater than it. We then show that an arbitrary subset of Endk⁢(V){\mathrm{End}_{k}(V)}is strictly triangularizable (defined in the obvious way) if and only if it is topologically nilpotent. This generalizes the theorem of Levitzki that every nilpotent semigroup of matrices is triangularizable. We also give a description of the triangularizable subalgebras of Endk⁢(V){\mathrm{End}_{k}(V)}, which generalizes a theorem of McCoy classifying triangularizable algebras of matrices over algebraically closed fields.

Journal

Forum Mathematicumde Gruyter

Published: Jan 1, 2019

References