Representations and the reduction theorem for ultragraph Leavitt path algebras

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.

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Correspondence to Daniel Gonçalves.

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Daniel Gonçalves: This author is partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) Grant Numbers 304487/2017-1 and 406122/2018-0 and Capes-PrInt Grant Number 88881.310538/2018-01 - Brazil.

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Gonçalves, D., Royer, D. Representations and the reduction theorem for ultragraph Leavitt path algebras. J Algebr Comb (2021). https://doi.org/10.1007/s10801-020-01004-8

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Mathematics Subject Classification

  • 16W50
  • 16S35
  • 16G99