2008
DOI: 10.1016/j.jalgebra.2008.05.020
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The classification question for Leavitt path algebras

Abstract: We prove an algebraic version of the Gauge-Invariant Uniqueness Theorem, a result which gives information about the injectivity of certain homomorphisms between Z-graded algebras. As our main application of this theorem, we obtain isomorphisms between the Leavitt path algebras of specified graphs. From these isomorphisms we are able to achieve two ends. First, we show that the K 0 groups of various sets of purely infinite simple Leavitt path algebras, together with the position of the identity element in K 0 ,… Show more

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Cited by 57 publications
(114 citation statements)
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“…Finally, our result demonstrates that data about the K 0 structure is sufficient to distinguish up to isomorphism the algebras in an important class of purely infinite simple K-algebras, thus paving a path for subsequent work by the authors [3] towards an algebraic version of [16], Theorem 4.2.4.…”
Section: Introductionmentioning
confidence: 67%
“…Finally, our result demonstrates that data about the K 0 structure is sufficient to distinguish up to isomorphism the algebras in an important class of purely infinite simple K-algebras, thus paving a path for subsequent work by the authors [3] towards an algebraic version of [16], Theorem 4.2.4.…”
Section: Introductionmentioning
confidence: 67%
“…(4) is independent of the left-right assumption. This shows that Gr-A is graded equivalent to Gr-B if and only if A-Gr is graded equivalent to B-Gr 2 Proposition 2.3.8 (3) shows that if all graded finitely generated projective Amodules are graded free, then Mod-A is graded equivalence to Mod-B if and only if B gr M n (A)(δ) for some n ∈ N and δ = (δ 1 , . .…”
Section: Theorem 238 Let a And B Be Two γ-Graded Rings The Followimentioning
confidence: 97%
“…Let us think about the "official" definition of the gauge action of a Leavitt path algebra B := L K (E) over the commutative (and unitary) ring K (see [1]). This is nothing but the group homomorphism τ : K × → aut(B) such that τ (z)(u) = u, τ (z)(f ) = zf and τ (z)(f * ) = z −1 f * for any vertex u, any edge f and any z ∈ K × .…”
Section: Drawbacks Of the Conventional Definitionmentioning
confidence: 99%
“…Since the notion of graded ideal and of gauge invariant ideal agree when we use the schematic version of the gauge action, such exceptionalities as the ones observed in [1,Proposition 1.7] are no longer present. As pointed out previously the gauge action in schematic sense is more demanding.…”
Section: Theoremmentioning
confidence: 99%