Using Steinberg algebras to study decomposability of Leavitt path\n algebras

2016/03/03 by Lisa Orloff Clark, Dolores Martı́n Barquero, Clark, Lisa Orloff +5
Mathematics · #06B10 #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Primary 16D70 #Rings and Algebras (math.RA) #Secondary 18B40

paper · pdf · doi:10.48550/arxiv.1603.01033

openalex publication_date 2016/03/03 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28

Abstract

Given an arbitrary graph E we investigate the relationship between E and\nthe groupoid G_E.\n We show that there is a lattice isomorphism between the lattice of pairs (H,\nS), where H is a hereditary and saturated set of vertices and S is a set\nof breaking vertices associated to H , onto the lattice of open invariant\nsubsets of G_E⁽⁰⁾. We use this lattice isomorphism to characterize the\ndecomposability of the Leavitt path algebra L_K(E), where K is a field.\n First we find a graph condition to characterise when an open invariant subset\nof G_E⁽⁰⁾ is closed.\n Then we give both a graph condition and a groupoid condition each of which is\nequivalent to L_K(E) being decomposable in the sense that it can be written\nas a direct sum of two nonzero ideals.\n

Related

• The groupoid approach to Leavitt path algebras
• A classification of ideals in Steinberg and Leavitt path algebras over\n arbitrary rings
• Classification of Leavitt path algebras with two vertices
• Leavitt path algebras of separated graphs
• Classification and Ideal Lattices of Leavitt Path Algebras