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

 Printed: Dec 2014
  • Leandro Cagliero,
    CIEM-CONICET, FAMAF-Universidad Nacional de Córdoba, Córdoba, Argentina.
  • Fernando Szechtman,
    Department of Mathematics and Statistics, University of Regina, Regina, SK
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We describe of 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 $F[x,y]=F[\alpha x+\beta y]$ for some non-zero elements $\alpha,\beta\in F$?
Keywords: uniserial module, Lie algebra, associative algebra, primitive element uniserial module, Lie algebra, associative algebra, primitive element
MSC Classifications: 17B10, 13C05, 12F10, 12E20 show english descriptions Representations, algebraic theory (weights)
Structure, classification theorems
Separable extensions, Galois theory
Finite fields (field-theoretic aspects)
17B10 - Representations, algebraic theory (weights)
13C05 - Structure, classification theorems
12F10 - Separable extensions, Galois theory
12E20 - Finite fields (field-theoretic aspects)

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