Leavitt Path Algebras

Abrams, Gene; Ara, Pere; Molina, Mercedes Siles

doi:10.1007/978-1-4471-7344-1

Cited by 221 publications

(591 citation statements)

References 16 publications

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“…, n. Using this representation, it is direct to see that L K (E) is a unital ring if and only if E 0 is finite in which case the sum of all vertices is the identity. For more details on these basic properties, see [1].…”

Section: Prerequisitesmentioning

confidence: 99%

Realization of graded matrix algebras as Leavitt path algebras

Vaš

2020

Beitr Algebra Geom

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While every matrix algebra over a field K can be realized as a Leavitt path algebra, this is not the case for every graded matrix algebra over a graded field. We provide a complete description of graded matrix algebras over a field, trivially graded by the ring of integers, which are graded isomorphic to Leavitt path algebras. As a consequence, we show that there are graded corners of Leavitt path algebras which are not graded isomorphic to Leavitt path algebras. This contrasts a recent result stating that every corner of a Leavitt path algebra of a finite graph is isomorphic to a Leavitt path algebra. If R is a finite direct sum of graded matricial algebras over a trivially graded field and over naturally graded fields of Laurent polynomials, we also present conditions under which R can be realized as a Leavitt path algebra.

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Section: Prerequisitesmentioning

confidence: 99%

Realization of graded matrix algebras as Leavitt path algebras

Vaš

2020

Beitr Algebra Geom

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“…1.3]). For more details about Cohn path algebras and Leavitt path algebras, we refer the reader to the monograph by Abrams, Ara and Siles Molina [1].…”

Section: Proof Of Theorem 14mentioning

confidence: 99%

“…For the rest of this section, we will reduce the general case of Theorem 1.4 to the finite case dealt with in Corollary 4.6. For Leavitt path algebras over a field, it is known that any Leavitt path algebra is the direct limit of Leavitt path algebras associated to finite graphs (see [1,Cor. 1.6.11]).…”

Section: Finite Graphsmentioning

confidence: 99%

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Chain Conditions for Epsilon-Strongly Graded Rings with Applications to Leavitt Path Algebras

Lännström

2019

Algebr Represent Theor

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We provide a characterization of graded von Neumann regular rings involving the recently introduced class of nearly epsilon-strongly graded rings. As our main application, we generalize Hazrat's result that Leavitt path algebras over fields are graded von Neumann regular. More precisely, we show that a Leavitt path algebra LR(E) with coefficients in a unital ring R is graded von Neumann regular if and only if R is von Neumann regular. We also prove that both Leavitt path algebras and corner skew Laurent polynomial rings over von Neumann regular rings are semiprimitive and semiprime. Thereby, we generalize a result by Abrams and Aranda Pino on the semiprimitivity of Leavitt path algebras over fields.

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“…The notion of a Cohn path algebra has been defined and investigated by Ara and Goodearl [5] (see also [2]). Specifically, for an arbitrary graph E = (E 0 , E 1 , s, r) and an arbitrary field K, the Cohn path algebra C K (E) of the graph E with coefficients in K is the K-algebra generated by the sets E 0 and E 1 , together with a set of variables {e * | e ∈ E 1 }, satisfying the following relations for all v, w ∈ E 0 and e, f ∈ E 1 :…”

Section: Introductionmentioning

confidence: 99%