Leavitt path algebras of separated graphs

By P. Ara, K. Goodearl. 2010

124 citations recorded by the source corpus.

Abstract The construction of the Leavitt path algebra associated to a directed graph E is extended to incorporate a family C consisting of partitions of the sets of edges emanating from the vertices of E. The new algebras, LK(E, C), are analyzed in terms of their homology, ideal theory, and K-theory. These algebras are proved to be hereditary, and it is shown that any conical abelian monoid occurs as the monoid 𝒱(LK(E, C)) of isomorphism classes of finitely generated projective modules over one of these algebras. The lattice of trace ideals of LK(E, C) is determined by graph-theoretic data, namely as a lattice of certain pairs consisting of a subset of E0 and a subset of C. Necessary conditions for 𝒱(LK(E, C)) to be a refinement monoid are developed, together with a construction that embeds (E, C) in a separated graph (E+, C+) such that 𝒱(LK(E+, C+)) has refinement.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/431831946046d7e2984b75751675d10ceab15da9

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In Algebra and Number Theory

1. Roozbeh Hazrat
2. Aidan Sims
3. P. Ara
4. G. Abrams
5. Huanhuan Li
6. R. Exel
7. M. Tomforde
8. Ben Steinberg

In Mathematical Physics

1. Roozbeh Hazrat
2. Aidan Sims
3. P. Ara
4. G. Abrams
5. Huanhuan Li
6. R. Exel
7. Ben Steinberg
8. M. Tomforde

Citations in the corpus

• Graded Steinberg algebras and their representations (2017, methodology): “…4.1], we write M E = lim M ( E (cid:48) , C (cid:48) , T (cid:48) ) , where E (cid:48) ranges over all the finite complete subgraphs of E and Applying [Ara and Goodearl 2012, Construction 5.3], we get that M ( E (cid:48) , C (cid:48) , T (cid:48) ) = M (cid:101) E for some finite graph (cid:101) E .”
• Dynamical systems associated to separated graphs, graph algebras, and paradoxical decompositions☆ (2012, methodology): “A wider class of graph algebras has been introduced in [8] and [7].”
• Leavitt path algebras: the first decade (2014, background): “• In the context of separated graphs, both Leavitt path algebras and graph C-algebras of these objects were introduced essentially simultaneously in the articles [30] (2011), and [31] (2012).”