Direct limits of graded matrix algebras

TL;DR

This paper classifies direct limits of finite-dimensional graded matrix algebras over algebraically closed fields, describing their graded K_0 groups and conditions for absorbing graded-division algebras, linking algebraic limits to $C^*$-algebra theory.

Contribution

It provides an explicit classification of graded matrix algebra limits and characterizes their K_0 groups and absorption properties, advancing the understanding of algebraic counterparts to $C^*$-algebras.

Findings

Explicit description of graded K_0 groups for direct limits

Conditions for absorption of graded-division algebras

Classification of graded matrix algebra limits

Abstract

The direct limit of finite-dimensional semisimple associative algebras arises as a purely algebraic counterpart to important $C^\ast$-algebras. In this paper, we classify direct limits of matrix algebras endowed with a grading by a finite abelian group over an algebraically closed field. In particular, we give an explicit description of the graded $K_0$ group of the direct limit of matrix algebras, and we provide conditions under which this limit absorbs graded-division algebras.