Realization of graded matrix algebras as Leavitt path algebras
Lia Vas
TL;DR
This work characterizes exactly which graded matrix algebras over a trivially graded field $K$ and over naturally graded fields $K[x^m, x^{-m}]$ arise as Leavitt path algebras, revealing a complete graded isomorphism criterion via finite acyclic (sinks) and finite comet (cycles) graph structures. It shows that a graded matrix algebra $\operatorname{M}_n(K)(\gamma_1,\dots,\gamma_n)$ is realizable as $L_K(E)$ precisely when it admits a representatives form $\operatorname{M}_n(K)(0,l_1(1),\dots,l_k(k))$ with $n=1+\sum l_i$, and similarly for $K[x^m,x^{-m}]$ with a comet-graph realization; the results extend to finite direct sums of such blocks. A key contribution is the identification of graded corners of LPAs that are not graded isomorphic to LPAs, highlighting intrinsic graded-structure obstructions absent in the ungraded setting. Together, these findings delineate the scope and limits of realizing graded matricial algebras as Leavitt path algebras and provide concrete criteria for when direct sums of such blocks can be realized by LPAs.
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.