Realization of graded matrix algebras as Leavitt path algebras
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Realization of graded matrix algebras as Leavitt path algebras Lia Vaš1 Received: 5 October 2019 / Accepted: 21 January 2020 © The Managing Editors 2020
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. Keywords Graded matrix algebra · Leavitt path algebra Mathematics Subject Classification 16W50 · 16S50 · 16D70
1 Introduction Every matrix algebra over a field K or the ring K [x, x −1 ] is isomorphic to a Leavitt path algebra. In contrast, not every graded matrix algebra over a field is graded isomorphic to a Leavitt path algebra by Vaš (2018, Proposition 3.7). Here, a Leavitt path algebra is considered with the natural grading by the ring of integers Z and the field K is considered to be trivially Z-graded. The Leavitt Path Algebra Realization Question of Vaš (2018, Section 3.3) asks for a characterization of those graded matrix algebras over K which can be realized as Leavitt path algebras. In Proposition 3.2, we answer this question by providing a complete characterization of graded matrix algebras over
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Lia Vaš [email protected] Department of Mathematics, Physics and Statistics, University of the Sciences, Philadelphia, PA 19104, USA
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Beitr Algebra Geom
K which are graded isomorphic to Leavitt path algebras. In Proposition 3.4, we provide analogous characterization for graded matrix algebras over naturally Z-graded K [x m , x −m ] for any positive integer m. These two results are used in Proposition 3.5 which presents conditions under which a finite direct sum of graded matricial algebras over K and K [x m , x −m ] can be realized by a Leavitt path algebra. As a consequence of Proposition 3.2, we show that there are graded corners of Leavitt path algebras which are not graded isomorphic to Leavitt path algebras (Example 3.6). This contrasts a recent result from Abrams and Nam (2020) which states that every corner of a Leavitt path algebra of a finite graph is isomorphic to another Leavitt path algebra.
2 Prerequisites A ring R is graded by a group if R = γ ∈ Rγ for additive subgroups Rγ and Rγ Rδ ⊆ Rγ δ for all γ , δ ∈ . The elements of the set H = γ ∈ Rγ are said to be homogeneous. The grading is trivial if Rγ = 0 for every nonidentity γ ∈ . A graded ring R is a graded division ring if every nonzero ho
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