scholarly journals The graded Grothendieck group and the classification of Leavitt path algebras

2012 ◽  
Vol 355 (1) ◽  
pp. 273-325 ◽  
Author(s):  
R. Hazrat
Keyword(s):  
Grothendieck Group ◽  
Path Algebras ◽  
Leavitt Path Algebras

scholarly journals K-theory classification of graded ultramatricial algebras with involution

2019 ◽  
Vol 31 (2) ◽  
pp. 419-463 ◽  
Author(s):  
Roozbeh Hazrat ◽  
Lia Vaš
Keyword(s):  
Grothendieck Group ◽  
Constructive Proof ◽  
Degree Zero ◽  
Formula One ◽  
Unitary Element ◽  
Path Algebras ◽  
Specific Formula ◽  
Involutive Structure ◽  
Leavitt Path Algebras

AbstractWe consider a generalization {K_{0}^{\operatorname{gr}}(R)} of the standard Grothendieck group {K_{0}(R)} of a graded ring R with involution. If Γ is an abelian group, we show that {K_{0}^{\operatorname{gr}}} completely classifies graded ultramatricial {*}-algebras over a Γ-graded {*}-field A such that (1) each nontrivial graded component of A has a unitary element in which case we say that A has enough unitaries, and (2) the zero-component {A_{0}} is 2-proper ({aa^{*}+bb^{*}=0} implies {a=b=0} for any {a,b\in A_{0}}) and {*}-pythagorean (for any {a,b\in A_{0}} one has {aa^{*}+bb^{*}=cc^{*}} for some {c\in A_{0}}). If the involutive structure is not considered, our result implies that {K_{0}^{\operatorname{gr}}} completely classifies graded ultramatricial algebras over any graded field A. If the grading is trivial and the involutive structure is not considered, we obtain some well-known results as corollaries. If R and S are graded matricial {*}-algebras over a Γ-graded {*}-field A with enough unitaries and {f:K_{0}^{\operatorname{gr}}(R)\to K_{0}^{\operatorname{gr}}(S)} is a contractive {\mathbb{Z}[\Gamma]}-module homomorphism, we present a specific formula for a graded {*}-homomorphism {\phi:R\to S} with {K_{0}^{\operatorname{gr}}(\phi)=f}. If the grading is trivial and the involutive structure is not considered, our constructive proof implies the known results with existential proofs. If {A_{0}} is 2-proper and {*}-pythagorean, we also show that two graded {*}-homomorphisms {\phi,\psi:R\to S} are such that {K_{0}^{\operatorname{gr}}(\phi)=K_{0}^{\operatorname{gr}}(\psi)} if and only if there is a unitary element u of degree zero in S such that {\phi(r)=u\psi(r)u^{*}} for any {r\in R}. As an application of our results, we show that the graded version of the Isomorphism Conjecture holds for a class of Leavitt path algebras: if E and F are countable, row-finite, no-exit graphs in which every infinite path ends in a sink or a cycle and K is a 2-proper and {*}-pythagorean field, then the Leavitt path algebras {L_{K}(E)} and {L_{K}(F)} are isomorphic as graded rings if any only if they are isomorphic as graded {*}-algebras. We also present examples which illustrate that {K_{0}^{\operatorname{gr}}} produces a finer invariant than {K_{0}}.

scholarly journals A classification of ideals in Steinberg and Leavitt path algebras over arbitrary rings

2021 ◽  
Vol 588 ◽  
pp. 200-249
Author(s):  
Simon W. Rigby ◽  
Thibaud van den Hove
Keyword(s):  
Path Algebras ◽  
Leavitt Path Algebras

scholarly journals Homotopy classification of Leavitt path algebras

2020 ◽  
Vol 362 ◽  
pp. 106961
Author(s):  
Guillermo Cortiñas ◽  
Diego Montero
Keyword(s):  
Homotopy Classification ◽  
Path Algebras ◽  
Leavitt Path Algebras

scholarly journals Flow invariants in the classification of Leavitt path algebras

2011 ◽  
Vol 333 (1) ◽  
pp. 202-231 ◽  
Author(s):  
Gene Abrams ◽  
Adel Louly ◽  
Enrique Pardo ◽  
Christopher Smith
Keyword(s):  
Path Algebras ◽  
Leavitt Path Algebras

scholarly journals Classification of Leavitt Path Algebras with Two Vertices

2019 ◽  
Vol 19 (3) ◽  
pp. 523-548
Author(s):  
Müge Kanuni ◽  
Dolores Martín Barquero ◽  
Cándido Martín González ◽  
Mercedes Siles Molina
Keyword(s):  
Path Algebras ◽  
Leavitt Path Algebras

scholarly journals On the representations of Leavitt path algebras

2011 ◽  
Vol 333 (1) ◽  
pp. 258-272 ◽  
Author(s):  
Daniel Gonçalves ◽  
Danilo Royer
Keyword(s):  
Path Algebras ◽  
Leavitt Path Algebras

scholarly journals FP-injective semirings, semigroup rings and Leavitt path algebras

2016 ◽  
Vol 45 (5) ◽  
pp. 1893-1906 ◽  
Author(s):  
Marianne Johnson ◽  
Tran Giang Nam
Keyword(s):  
Semigroup Rings ◽  
Path Algebras ◽  
Leavitt Path Algebras
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