On the Theorem of the Primitive Element with Applications to the Representation Theory of Associative and Lie Algebras. Issue 4 (1st December 2014)

• Record Type: Journal Article
• Authors: Cagliero, Leandro; Szechtman, Fernando
• Is Part Of: Canadian mathematical bulletin =. Volume 57:Issue 4(2014)
• Page Start: 735
• Page End: 748
• Publication Date: 2014-12-01
• Subjects: 17B10 -- 13C05 -- 12F10 -- 12E20; uniserial module -- Lie algebra -- associative algebra -- primitive element; Mathematics -- Periodicals; Mathematics; Periodicals; 510.5
• DOI: 10.4153/CMB-2013-046-9
• Languages: English
• ISSNs: 0008-4395

Abstract

Abstract: 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 $\alpha, \, \beta \, \in \, F?$