On some relationships between Leavitt path algebras and Brauer configuration algebras

Abstract

We compare Cuntz–Krieger and Brauer relations inside the same path algebra when every configuration vertex has valency two. Choosing one arrow from each successor pair produces a directed graph, and the Cuntz–Krieger quotient is its Leavitt path algebra. The Brauer quotient is the usual Brauer configuration algebra; it belongs to the Brauer graph class when all polygons are 2-gons. We also determine the two mixed quotients. One can be described for relaxed data and disappears under the standard Brauer axioms, while the other is always zero. In the converse direction, any finite directed graph can be recovered from a standard Brauer configuration and a cut, with isolated vertices accounting for scalar summands. The choice of cut affects more than growth: for diamond-tail and complete binary-tree configurations it changes the Grothendieck group. In the balanced cases we obtain closed spectral formulas and a determinant recurrence for the orders of the finite groups that occur.