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On the Theorem of the Primitive Element with Applications to the Representation Theory of Associative and Lie Algebras

Published online by Cambridge University Press:  20 November 2018

Leandro Cagliero  and
Fernando Szechtman
Leandro Cagliero
Affiliation:
CIEM-CONICET, FAMAF-Universidad Nacional de Córdoba, Cördoba, Argentina. e-mail: [email protected]
Fernando Szechtman
Affiliation:
Department of Mathematics and Statistics, University of Regina, Regina, SK e-mail: [email protected]
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Abstract

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We describe all finite dimensional uniserial representations of a commutative associative (resp. abelian Lie) algebra over a perfect (resp. sufficiently large perfect) field. In the Lie case the size of the field depends on the answer to following question, considered and solved in this paper. Let $K/F$ be a finite separable field extension and let $x,\,y\,\in \,K$. When is $F\left[ x,\,y \right]\,=\,F\left[ \alpha x\,+\,\beta y \right]$ for some nonzero elements $\alpha ,\,\beta \,\in \,F?$

Keywords

17B1013C0512F1012E20uniserial moduleLie algebraassociative algebraprimitive element
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 57 , Issue 4 , 01 December 2014 , pp. 735 - 748
Copyright
Copyright © Canadian Mathematical Society 2014

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