On Lie and associative algebras containing inner derivations

TL;DR

This paper characterizes subalgebras of the Lie algebra containing all inner derivations of matrix algebras and proves a density theorem for associative algebras generated by these derivations.

Contribution

It provides a detailed description of subalgebras containing inner derivations and establishes a density theorem in a broader algebraic context.

Findings

Characterization of subalgebras of (n^2) containing all inner derivations.

Proof of a density theorem for associative algebras generated by inner derivations.

Extension of results to prime algebras under certain conditions.

Abstract

We describe subalgebras of the Lie algebra $\mf{gl}(n^2)$ that contain all inner derivations of $A=M_n(F)$ (where $n\ge 5$ and $F$ is an algebraically closed field of characteristic 0). In a more general context where $A$ is a prime algebra satisfying certain technical restrictions, we establish a density theorem for the associative algebra generated by all inner derivations of $A$.