# Representations and the reduction theorem for ultragraph Leavitt path algebras

## TL;DR

This paper investigates ultragraph Leavitt path algebras by exploring their representations through branching systems, establishing a reduction theorem, and analyzing the conditions under which these representations are faithful or permutative.

## Contribution

It introduces a reduction theorem for ultragraph Leavitt path algebras and characterizes faithfulness and permutative representations in terms of branching system dynamics.

## Key findings

- Proved the reduction theorem for ultragraph Leavitt path algebras.
- Showed ultragraph Leavitt path algebras are semiprime.
- Provided criteria for permutative representations to be equivalent to branching system representations.

## Abstract

In this paper we study representations of ultragraph Leavitt path algebras via branching systems and, using partial skew ring theory, prove the reduction theorem for these algebras. We apply the reduction theorem to show that ultragraph Leavitt path algebras are semiprime and to completely describe faithfulness of the representations arising from branching systems, in terms of the dynamics of the branching systems. Furthermore, we study permutative representations and provide a sufficient criteria for a permutative representation of an ultragraph Leavitt path algebra to be equivalent to a representation arising from a branching system. We apply this criteria to describe a class of ultragraphs for which every representation (satisfying a mild condition) is permutative and has a restriction that is equivalent to a representation arising from a branching system.