A note on additive commutator groups in certain algebras

TL;DR

This paper investigates the structure of certain algebras, specifically whether they can be decomposed into their center plus additive commutators, with applications to matrix rings, quaternion algebras, and twisted group algebras.

Contribution

It provides new insights into the decomposition of unital associative algebras into centers and commutators, especially for matrix rings, quaternion algebras, and semisimple algebras.

Findings

Decomposition $A=Z(A)+[A,A]$ holds in specific algebra classes.

If all additive commutators are central in a twisted group algebra, then the algebra is commutative.

Characterization of when such decompositions are possible in various algebraic structures.

Abstract

We study whether a unital associative algebra $ A $ over a field admits a decomposition of the form $A = Z(A) + [A,A]$ where $ Z(A) $ is the center of $ A $ and $ [A,A] $ denotes the additive subgroup of $A$ generated by all additive commutators of $A$. Among our main considerations are the cases in which $A$ is the matrix ring over a division ring, a generalized quaternion algebra, or a semisimple finite-dimensional algebra. We also discuss some applications that do not necessarily require the decomposition, such as the case where $ A $ is the twisted group algebra of a locally finite group over a field of characteristic zero: if all additive commutators of $A$ are central, then $ A $ must be commutative.