Commutators on Generalized Block-Triangular Algebras
Pedro Souza Fagundes, Thiago Castilho de Mello
TL;DR
The paper addresses the classical problem of characterizing commutators in finite-dimensional algebras by introducing the multitrace invariant from the Wedderburn-Malcev decomposition $A=B\oplus\operatorname{rad}(A)$ with $B\cong M_{d_1}(K)\oplus\cdots\oplus M_{d_r}(K)$. It proves that in the generalized block-triangular setting, an element is a commutator if and only if its multitrace vanishes, extending the Albert-Muckenhoupt-Shoda criterion beyond full matrix algebras. The authors develop a module-theoretic Sylvester-Rosenblum equation to control off-diagonal interactions and use it inductively on the number of simple components to establish the main result, including a consequence that the sum of commutators remains a commutator. They also show that the radical is contained in the commutator set and obtain sharp dimension bounds for A/[A,A], with implications for polynomial images and related conjectures. Overall, the work provides a concrete, computable criterion for commutators in a broad algebra class and deepens understanding of how trace-type invariants govern commutator structure in finite-dimensional algebras.
Abstract
The characterization of commutators in associative algebras is a classical problem in ring theory. In this paper, we address this problem for the natural class of generalized block-triangular algebras. To this end, we introduce a new invariant: the multitrace of an arbitrary element in an associative unital algebra, and prove that in a generalized block-triangular algebra, an element is a commutator if and only if its multitrace vanishes. As a consequence, we show that the set of commutators is closed under addition in these algebras. Our main result extends the classical Albert-Muckenhoupt-Shoda theorem for full matrix algebras to the broader setting of generalized block-triangular algebras.