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.
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Notes
If M is a graded right R-module and \(\gamma \in \Gamma ,\) the \(\gamma \)-shifted or \(\gamma \)-suspended graded right R-module \((\gamma )M\) is defined as the module M with the \(\Gamma \)-grading given by
$$\begin{aligned} (\gamma )M_\delta = M_{\gamma \delta } \end{aligned}$$for all \(\delta \in \Gamma .\) Any finitely generated graded free right R-module is of the form \((\gamma _1)R\oplus \cdots \oplus (\gamma _n)R\) for \(\gamma _1, \ldots ,\gamma _n\in \Gamma \) and \({\text {Hom}}_R(F,F)\) is a \(\Gamma \)-graded ring which is graded isomorphic to \({\mathbb {M}}_n(R)(\gamma _1,\dots ,\gamma _n)\) (both Năstăsescu and van Oystaeyen 2004; Hazrat 2016 contain details).
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Vaš, L. Realization of graded matrix algebras as Leavitt path algebras. Beitr Algebra Geom 61, 771–781 (2020). https://doi.org/10.1007/s13366-020-00487-7
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DOI: https://doi.org/10.1007/s13366-020-00487-7