Algebraic characterisations of path algebras

Abstract

The theory of path algebras is usually circunscripted to the study of representations, usually linked to finite graphs. In our work, we focus on studying the structure of path algebras over a field associated to arbitrary graphs. We characterise perfection (simplicity, primitivity, primeness and semiprimeness) and finitness conditions (artinianity, semiartinianity and noetherianity) in terms of geometric conditions in the associated graph. In order to do so, we also compute the socle and the Jacobson radical of a path algebra. In addition, we study the centroid of any path algebra and the extended centroid and central closure of the path algebra of a cycle. We obtain two structure theorems, one for semiprime path algebras, and another for noetherian ones. Semiprime path algebras are direct sum of simple, prime and primitive algebras, and noetherian path algebras modulo its radical will be isomorphic to upper triangular formal matrix algebras, they can also be seen as direct sums of path algebras of cycles and copies of the ground field itself.