The dynamical structure of partial group algebras with relations, with applications to subshift algebras

TL;DR

This paper develops a framework for partial group algebras with relations, linking algebraic and topological partial actions, and applies it to analyze the structure and simplicity of subshift algebras.

Contribution

It introduces a new algebraic framework for partial group algebras with relations and connects it to topological partial actions, with applications to subshift algebra analysis.

Findings

Partial skew group rings are isomorphic to partial group algebras with relations.

Under certain conditions, partial skew group rings can be described via topological partial actions.

Simplicity of subshift algebras is characterized by the dynamics of the underlying subshift.

Abstract

We introduce partial group algebras with relations in a purely algebraic framework. Given a group and a set of relations, we define an algebraic partial action and prove that the resulting partial skew group ring is isomorphic to the associated partial group algebra with relations. Under suitable conditions - which always holds if the base ring is a field - we demonstrate that the partial skew group ring can also be described using a topological partial action. Furthermore, we show how subshift algebras can be realized as partial group algebras with relations. Using the topological partial action, we describe simplicity of subshift algebras in terms of the underlying dynamics of the subshift.