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

TL;DR

This paper characterizes all finite-dimensional uniserial representations of commutative associative and abelian Lie algebras over certain fields, linking the structure to a key field extension problem involving elements and their generated subalgebras.

Contribution

It provides a complete description of uniserial representations for these algebras and solves a fundamental field extension problem crucial for the Lie case.

Findings

Classification of uniserial representations over perfect fields

Solution to the field extension problem involving elements and subalgebras

Conditions under which certain subalgebras are generated by linear combinations

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 $x,y\in K$. When is $F[x,y]=F[\alpha x+\beta y]$ for some non-zero elements $\alpha,\beta\in F$?