Structure and characterizations of algebras associated with graphs.

Abstract

Algebras related to graphs such as path, Leavitt path and Steinberg algebras arise from different contexts yet remain closely connected. All three are constructed from directed graphs, where algebraic and geometric properties are deeply intertwined. Path algebras naturally embed into Leavitt path algebras, which can also be realized as quotients of the former, while Steinberg algebras provide a broader framework that extends Leavitt path algebras.

Although Leavitt and Steinberg algebras are mentioned, this work focuses primarily on path algebras. Within this framework, we examine several of their structural and algebraic properties, including Noetherianity, Artinianity, the Jacobson radical, the socle chain, among others. We also introduce a new measure for filtered algebras: algebraic entropy. Originally developed in physics and later adapted to areas such as information theory, dynamical systems and group theory. Entropy was more recently extended to graded algebras, which motivates the development of a new version suitable for filtered algebras. We then study the behavior of this new notion of entropy under various transformations, such as direct sums, monomorphism and epimorphisms, In the context of path and Leavitt path algebras, we prove that the entropy of the path algebra with the standard filtration of a finite graph fall within on of the cases of the trichotomy theorem, and furthermore, we show that the entropy of path and Leavitt path algebras coincide. Finally, in the last past of the manuscript, we explore field-independence phenomena, making use of category theory and extensions of the Hilbert's Nullstellensatz.