On the theorem of the primitive element with applications to the representation theory of associative and Lie algebras

2013/06/17 by Leandro Cagliero, Cagliero, Leandro, Fernando Szechtman +1
Mathematics · #12E20 #13C05 #Abelian group #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebra over a field #Algebra representation #Algebraic structures and combinatorial models #Associative algebra #Associative property #Commutative property #Computer science #Division algebra #Element (criminal law) #Extension (predicate logic) #FOS: Mathematics #Field (mathematics) #Lie algebra #Mathematical analysis #Mathematics #Primary 17B10 #Pure mathematics #Representation Theory (math.RT) #Representation theory #Secondary 12F10 #Separable space #Zero (linguistics) #math.RT #msc:12E20 #msc:12F10 #msc:13C05 #msc:17B10

paper · pdf · doi:10.48550/arxiv.1306.3965

arxiv created 2013/06/17 · openalex publication_date 2013/06/17 · arxiv updated 2013/06/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06

Abstract

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∈ K. When is F[x,y]=F[αx+βy] for some non-zero elements α,β∈ F?

Citations

• On wild Lie algebras
• Algebraische Theorie der Körper.
• Representation Theory of Artin Algebras
• Generalized Artin-Schreier polynomials

Related

• On the number of generators of a separable algebra over a finite field
• Universal Lie algebra extensions via commutative structures
• Partial Representations and Partial Group Algebras
• Primitive ideals, non-restricted representations and finite W-algebras
• Primitive Element Theorem for Fields with Commuting Derivations and Automorphisms