Project's information

Project's title Investigation of Leavitt path algebras partially by methods from commutative algebras and groups
Project’s code QTHU01.02-2021
Research hosting institution Institute of Mathematics
Project leader’s name Tran Giang Nam
Project duration 01/06/2020 - 31/12/2022
Project’s budget 200 million VND
Classify Excellent
Goal and objectives of the project

The main goal is to study some open questions of  Leavitt path algebras and graph C*-algebras, by methods from commutative algebras and groups. The main problems to be considered are:
Problem 1: Classify Leavitt path algebras and C*-algebras via K-theory and KK-theory.
Problem 2: Find criteria for two Leavitt path algebras to be isomorphic.
Problem 3: Find criteria for irreducible representations of Leavitt path algebras via their diagonal subalgebras. 
Main results

- Theorerical results: For Problem 1, we give a survey about the classification of Leavitt path algebras and C*-algebras via K-theory and KK-theory.
   For Problem 2, we give a survey about the criteria for two Leavitt path algebras to be isomorphic.
   For Problem 3, we give many ways to construct irreducible representions of Leavitt path algebras associted with vertices via Cohn Leavitt path algebras, and give a universal method to construct irreducible representions of Leavitt path algebras associted pairs (c, f) consisting of simple closed paths c together with irreducible polynomials f. We compute directly the kernel of these irreducible representations without using the theory of graded ideals of Tomforde. In particular, we give a new method to construct irreducible representions of Leavitt path algebras via their diagonal subalgebras.
- Applied results: Two PhD students were supported by this project to make two preprints.
Novelty and actuality and scientific meaningfulness of the results

We give many ways to construct irreducible representions of Leavitt path algebras associted with vertices via Cohn Leavitt path algebras, and give a universal method to construct irreducible representions of Leavitt path algebras associted pairs (c, f) consisting of simple closed paths c together with irreducible polynomials f. In particular, we give a new method to construct irreducible representions of Leavitt path algebras via their diagonal subalgebras.
Products of the project

- Scientific papers in referred journals (list):
   1. P. N. Ánh and T. G. Nam, Special irreducible representations of Leavitt path algebras, Advances in Mathematics 377 (2021) 107483.
- Other products (if applicable): A summary of the goals and results of the project.