Document Zbl 1127.16011

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Math.</a> <a href="/?q=in%3A185189" title="Articles in this Issue">204, No. 2, 572-618 (2006)</a>. </div> <div class="abstract">From the authors’ summary: We introduce a new category $\mathcal C$, which we call the cluster category, obtained as a quotient of the bounded derived category $\mathcal D$ of the module category of a finite-dimensional hereditary algebra $H$ over a field. We show that, in the simply laced Dynkin case, $\mathcal C$ can be regarded as a natural model for the combinatorics of the corresponding Fomin-Zelevinsky cluster algebra. Using approximation theory, we investigate the tilting theory of $\mathcal C$, showing that it is more regular than that of the module category itself, and demonstrating an interesting link with the classification of self-injective algebras of finite representation type. This investigation also enables us to conjecture a generalisation of APR-tilting.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A6977">Xi Changchang (Beijing)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A5047155">Cited in <strong>41</strong> Reviews</a></div><div class="clear"><a href="/?q=rf%3A5047155">Cited in <strong>458</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A16G20" title="MSC2020">16G20</a> </td> <td class="space"> Representations of quivers and partially ordered sets </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A16G70" title="MSC2020">16G70</a> </td> <td class="space"> Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers </td> </tr><tr> <td class="mono"> 18E30 </td> <td class="space"> Derived categories, triangulated categories (MSC2010) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A16D90" title="MSC2020">16D90</a> </td> <td class="space"> Module categories in associative algebras </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Atilting+modules">tilting modules</a>; <a href="/?q=ut%3Acluster+algebras">cluster algebras</a>; <a href="/?q=ut%3Acluster+categories">cluster categories</a>; <a href="/?q=ut%3Abounded+derived+categories">bounded derived categories</a>; <a href="/?q=ut%3Amodule+categories+of+finite-dimensional+hereditary+algebras">module categories of finite-dimensional hereditary algebras</a>; <a href="/?q=ut%3Aapproximations">approximations</a>; <a href="/?q=ut%3Arepresentation+types">representation types</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1127.16011" data-ciurl="/ci/05047155" data-biburl="/bibtex/05047155.bib" data-amsurl="/amsrefs/05047155.bib" data-xmlurl="/xml/05047155.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/05047155.pdf" title="Zbl 1127.16011 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1016/j.aim.2005.06.003" aria-label="DOI for “Tilting theory and cluster combinatorics.”" title="10.1016/j.aim.2005.06.003">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/math/0402054"title="Note: arXiv document may differ from published version">arXiv</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Auslander, M.; Platzeck, M. 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