2020/03/22 by Luiz Gustavo Cordeiro, Daniel Gonçalves, Cordeiro, Luiz Gustavo +3 · 1 citation
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebraic structures and combinatorial models
paper · pdf · doi:10.48550/arxiv.2003.09911
It is a conjecture that for the class of Leavitt path algebras associated to\nfinite directed graphs, their graded Grothendieck groups K0\gr\nare a complete invariant. For a Leavitt path algebra L mathsf k(E), with\ncoefficient in a field mathsf k, the monoid of the positive cone of\nK0\gr(L mathsf k(E)) can be described completely in terms of\nthe graph E. In this note we further investigate the structure of this\n"talented monoid", showing how it captures intrinsic properties of the graph\nand hence the structure of its associated Leavitt path algebras. In particular,\nfor the class of strongly connected graphs, we show that the notion of the\nperiod of a graph can be completely described via the talented monoid. As an\napplication, we will give a finer characterisation of the purely infinite\nsimple Leavitt path algebras in terms of properties of the associated graph. We\nshow that graded isomorphism of algebras preserve the period of the graphs, and\nobtain results giving more evidence to the graded classification conjecture.\n