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Leavitt Path Algebras and Classical K-Theory pp 3–20Cite as

A Survey of Some of the Recent Developments in Leavitt Path Algebras

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Abstract

In this survey article, we describe some of recent ring-theoretic and module-theoretic investigations of a Leavitt path algebra L of an arbitrary directed graph E over a field K. It is shown how a single graph-theoretical property of E often gives rise to several independent ring properties of L, thus making Leavitt path algebras as effective tools in constructing examples of rings with various desired properties. Leavitt path algebras satisfying a polynomial identity are completely described. It is shown how using special vertices, infinite paths or cycles in the graph E, various types of simple modules over L can be constructed. A complete description is given of a Leavitt path algebra L whose simple modules possess various specific properties such as being, flat, injective, graded or finitely presented. In the first three cases, L becomes von Neumann regular while in the last case, when the graph E is finite, L possesses finite GK-dimension. Leavitt path algebras having only finitely many isomorphism classes of simple modules turn out to be semi-artinian von Neumann regular rings in which the ideals form a finite chain under inclusion. The sum and the intersection of any two principal one-sided ideals of L are shown to be again principal one-sided ideals and this leads to the existence of the left/right gcd and the left/right lcm of any two non-zero elements in L.

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Authors and Affiliations

  1. Department of Mathematics, University of Colorado, Colorado Springs, CO, 80918, USA

    Kulumani M. Rangaswamy

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  1. Kulumani M. Rangaswamy
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Correspondence to Kulumani M. Rangaswamy .

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Editors and Affiliations

  1. Department of Mathematics, Cochin University of Science and Technology, Cochin, Kerala, India

    A. A. Ambily

  2. Centre for Research in Mathematics, Western Sydney University, Sydney, NSW, Australia

    Roozbeh Hazrat

  3. Statistics and Mathematics Unit, Indian Statistical Institute, Bengaluru, Karnataka, India

    B. Sury

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Rangaswamy, K.M. (2020). A Survey of Some of the Recent Developments in Leavitt Path Algebras. In: Ambily, A., Hazrat, R., Sury, B. (eds) Leavitt Path Algebras and Classical K-Theory. Indian Statistical Institute Series. Springer, Singapore. https://doi.org/10.1007/978-981-15-1611-5_1

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