Realization of graded matrix algebras as Leavitt path algebras

TL;DR

This paper characterizes which graded matrix algebras over a field are isomorphic to Leavitt path algebras, revealing limitations and conditions for such realizations, especially in graded settings.

Contribution

It provides a complete description of graded matrix algebras over a field that are isomorphic to Leavitt path algebras and explores graded corners and realizability conditions.

Findings

Not all graded matrix algebras over a field are Leavitt path algebras.

Some graded corners of Leavitt path algebras are not isomorphic to Leavitt path algebras.

Conditions are given for realizing certain graded algebras as Leavitt path algebras.

Abstract

While every matrix algebra over a field $K$ can be realized as a Leavitt path algebra, this is not the case for every graded matrix algebra over a graded field. We provide a complete description of graded matrix algebras over a field, trivially graded by the ring of integers, which are graded isomorphic to Leavitt path algebras. As a consequence, we show that there are graded corners of Leavitt path algebras which are not graded isomorphic to Leavitt path algebras. This contrasts a recent result stating that every corner of a Leavitt path algebra of a finite graph is isomorphic to a Leavitt path algebra. If $R$ is a finite direct sum of graded matricial algebras over a trivially graded field and over naturally graded fields of Laurent polynomials, we also present conditions under which $R$ can be realized as a Leavitt path algebra.