Regularity Conditions for Arbitrary Leavitt Path Algebras

By G. Abrams, Kulumani M. Rangaswamy. 2008

73 citations recorded by the source corpus.

We show that if E is an arbitrary acyclic graph then the Leavitt path algebra LK(E) is locally K-matricial; that is, LK(E) is the direct union of subalgebras, each isomorphic to a finite direct sum of finite matrix rings over the field K. (Here an arbitrary graph means that neither cardinality conditions nor graph-theoretic conditions (e.g. row-finiteness) are imposed on E. These unrestrictive conditions are in contrast to the hypotheses used in much of the literature on this subject.) As a consequence we get our main result, in which we show that the following conditions are equivalent for an arbitrary graph E: (1) LK(E) is von Neumann regular. (2) LK(E) is π-regular. (3) E is acyclic. (4) LK(E) is locally K-matricial. (5) LK(E) is strongly π-regular. We conclude by showing how additional regularity conditions (unit regularity, strongly clean) can be appended to this list of equivalent conditions.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/83e1e4d1167a745ef339d670a41173d47e6031b1

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In Algebra and Number Theory

1. G. Abrams
2. M. Tomforde
3. Kulumani M. Rangaswamy
4. G. Pino
5. P. Ara
6. Mercedes Siles Molina
7. Roozbeh Hazrat
8. Aidan Sims

In Computational Theory and Mathematics

1. G. Abrams
2. Kulumani M. Rangaswamy
3. M. Tomforde

In Mathematical Physics

1. G. Abrams
2. M. Tomforde
3. Kulumani M. Rangaswamy
4. G. Pino
5. P. Ara
6. Mercedes Siles Molina
7. Roozbeh Hazrat
8. Aidan Sims

Citations in the corpus

• The Theory of Prime Ideals of Leavitt Path Algebras over Arbitrary Graphs (2011, background): “Properties of LK(Pκ): The Leavitt path algebra LK(Pκ) of the Pyramid graph Pκ has many interesting properties: (i) It is a primitive von Neumann regular ring (as E is acyclic; see [8]) (ii) Every ideal of LK(Pκ) is a primitive ideal and is further graded (Justification: E is acyclic, E satisfies the”
• Isomorphism and Morita equivalence of graph algebras (2008, background): “The result is also shown directly, for arbitrary sized graphs, in [10].”
• Leavitt path algebras: the first decade (2014, background): “It was shown in [16] that the only von Neumann regular Leavitt path algebras are those associated to acyclic graphs, so it would initially seem that Leavitt path algebras would not be fertile ground in the context of the Realization Problem.”