On theorems of Brauer-Nesbitt and Brandt for characterizations of small block algebras

TL;DR

This paper generalizes classical theorems by Brauer-Nesbitt and Brandt, characterizing small block algebras through the codimension of their commutator subspace, extending results to all finite-dimensional algebras.

Contribution

It redefines the invariant k(A) as the codimension of the commutator subspace and extends the characterization to arbitrary finite-dimensional algebras, not just symmetric ones.

Findings

Proves Morita invariance of the codimension of the sum of K(A) and Rad^n(A).

Provides an upper bound for the codimension.

Extends Okuyama's refinement to non-symmetric algebras.

Abstract

In 1941, Brauer-Nesbitt established a characterization of a block with trivial defect group as a block $B$ with $k(B) = 1$ where $k(B)$ is the number of irreducible ordinary characters of $B$. In 1982, Brandt established a characterization of a block with defect group of order two as a block $B$ with $k(B) = 2$. These correspond to the cases when the block is Morita equivalent to the one-dimensional algebra and to the non-semisimple two-dimensional algebra, respectively. In this paper, we redefine $k(A)$ to be the codimension of the commutator subspace $K(A)$ of a finite-dimensional algebra $A$ and prove analogous statements for arbitrary (not necessarily symmetric) finite-dimensional algebras. This is achieved by extending the Okuyama refinement of the Brandt result to this setting. To this end, we study the codimension of the sum of the commutator subspace $K(A)$ and $n$th Jacobson radical $\operatorname{Rad}^n(A)$. We prove that this is Morita invariant and give an upper bound for the codimension as well.