Theorems of Burnside and Wedderburn revisited

TL;DR

This paper revisits classical theorems of Burnside and Wedderburn through the lens of simultaneous triangularization, providing new characterizations and extensions for matrix algebras over various fields.

Contribution

It generalizes Burnside's theorem for irreducible subalgebras of triangularizable matrices and extends Wedderburn's theorem to certain nilpotent subalgebras over division rings.

Findings

Characterization of fields where Burnside's theorem holds.

Extension of Burnside's theorem to subalgebras of division rings.

Extension of Wedderburn's theorem for nilpotent subalgebras.

Abstract

We approach celebrated theorems of Burnside and Wedderburn via simultaneous triangularization. First, for a general field $F$, we prove that $M_n(F)$ is the only irrreducible subalgebra of triangularizable matrices in $M_n(F)$ provided such a subalgebra exists. This provides a slight generalization of a well-known theorem of Burnside. Next, for a given $n > 1$, we characterize all fields $F$ such that Burnside's Theorem holds in $M_n(F)$, i.e., $M_n(F)$ is the only irreducible subalgebra of itself. In fact, for a subfield $F$ of the center of a division ring $D$, our simple proof of the aforementioned extension of Burnside's Theorem can be adjusted to establish a Burnside type theorem for irreducible $F$-algebras of triangularizable matrices in $M_n(D)$ with inner eigenvalues in $F$, namely such subalgebras of $M_n(D)$ are similar to $M_n(F)$. We use Burnside's theorem to present a simple proof of a theorem due to Wedderburn. Then, we use our Burnside type theorem to prove an extension of Wedderburn's Theorem as follows: A subalgebra of a semi-simple left Artinian $F$-algebra is nilpotent iff the algebra, as a vector space over the field $F$, is spanned by its nilpotent members and that the minimal polynomials of all of its members split into linear factors over $F$. We conclude with an application of Wedderburn's Theorem.