On small matrix subalgebras with a trivial centralizer

Abstract

Given an integer n ≥ 3, we investigate the minimal dimension of a subalgebra of M _n (𝕂) with a trivial centralizer. It is shown that this dimension is 5 when n is even and 4 when it is odd. In the latter case, we also determine all 4-dimensional subalgebras with a trivial centralizer.

Keywords:

• commutation
• matrices
• algebras
• dimension
• centralizer

AMS Subject Classifications:

• 15A45
• 16S50

Acknowledgements

I would like to thank Saab Abou-Jaoudé for submitting the original problem to me and Pierre Mazet for inspiring a part of this work.