{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T06:44:34Z","timestamp":1740120274289,"version":"3.37.3"},"reference-count":14,"publisher":"World Scientific Pub Co Pte Ltd","issue":"08","funder":[{"DOI":"10.13039\/501100000923","name":"Australian Research Council","doi-asserted-by":"crossref","award":["DP160101481"],"award-info":[{"award-number":["DP160101481"]}],"id":[{"id":"10.13039\/501100000923","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Algebra Comput."],"published-print":{"date-parts":[[2021,12]]},"abstract":"<jats:p> If [Formula: see text] is a directed graph and [Formula: see text] is a field, the Leavitt path algebra [Formula: see text] of [Formula: see text] over [Formula: see text] is naturally graded by the group of integers [Formula: see text] We formulate properties of the graph [Formula: see text] which are equivalent with [Formula: see text] being a crossed product, a skew group ring, or a group ring with respect to this natural grading. We state this main result so that the algebra properties of [Formula: see text] are also characterized in terms of the pre-ordered group properties of the Grothendieck [Formula: see text]-group of [Formula: see text]. If [Formula: see text] has finitely many vertices, we characterize when [Formula: see text] is strongly graded in terms of the properties of [Formula: see text] Our proof also provides an alternative to the known proof of the equivalence [Formula: see text] is strongly graded if and only if [Formula: see text] has no sinks for a finite graph [Formula: see text] We also show that, if unital, the algebra [Formula: see text] is strongly graded and graded unit-regular if and only if [Formula: see text] is a crossed product. <\/jats:p><jats:p> In the process of showing the main result, we obtain conditions on a group [Formula: see text] and a [Formula: see text]-graded division ring [Formula: see text] equivalent with the requirements that a [Formula: see text]-graded matrix ring [Formula: see text] over [Formula: see text] is strongly graded, a crossed product, a skew group ring, or a group ring. We characterize these properties also in terms of the action of the group [Formula: see text] on the Grothendieck [Formula: see text]-group [Formula: see text] <\/jats:p>","DOI":"10.1142\/s0218196722500102","type":"journal-article","created":{"date-parts":[[2021,11,25]],"date-time":"2021-11-25T09:28:34Z","timestamp":1637832514000},"page":"1753-1773","source":"Crossref","is-referenced-by-count":0,"title":["Crossed product Leavitt path algebras"],"prefix":"10.1142","volume":"31","author":[{"given":"Roozbeh","family":"Hazrat","sequence":"first","affiliation":[{"name":"Centre for Research in Mathematics and Data Science, Western Sydney University, Australia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lia","family":"Va\u0161","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Physics and Statistics, University of the Sciences, Philadelphia, PA 19104, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2021,11,25]]},"reference":[{"key":"S0218196722500102BIB001","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4471-7344-1"},{"key":"S0218196722500102BIB002","first-page":"165","volume":"669","author":"Ara P.","year":"2012","journal-title":"J. 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