The maximal dimension of unital subalgebras of the matrix algebra

Abstract

Using Wederburn's main theorem and a result of Gerstenhaber we prove that, over a field of characteristic zero, the maximal dimension of a proper unital subalgebra in the $n\times n$ matrix algebra is $n^2-n+1$ and furthermore this upper bound is attained for the so-called parabolic subalgebras. We also investigate the corresponding notion of parabolic coideals for matrix coalgebras and prove that the minimal dimension of a non-zero coideal of the matrix coalgebra ${\mathcal{M}}^n(k)$ is $n-1$.

7 pages; title changed. Final version to appear in Forum Math. DOI:10.1515/forum-2015-0241

References in corpus (3)

• Extending structures, Galois groups and supersolvable associative algebras
• On the dimensions of commutative subalgebras and subgroups
• On abelian subalgebras and ideals of maximal dimension in supersolvable lie algebras

Cited by in corpus (2)

• Asymptotic of Number of Similarity Classes of Commuting Tuples
• On the codimension of subalgebras of the algebra of matrices over a field