<p>We describe the centroid of some Leavitt path algebras. More precisely, for Leavitt path algebras over a field <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10801_2025_1414_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">${{\mathbb {K}}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">K</mi> </math></EquationSource> </InlineEquation>, we show that if the algebra is simple, then its centroid is isomorphic to <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10801_2025_1414_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">${{\mathbb {K}}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">K</mi> </math></EquationSource> </InlineEquation>, and if the algebra is prime, then its centroid is also isomorphic to <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10801_2025_1414_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">${{\mathbb {K}}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">K</mi> </math></EquationSource> </InlineEquation>, except if the graph is a row-finite comet, in which case the centroid is isomorphic to <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10801_2025_1414_Article_IEq4.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="63" /> </InlineMediaObject> <EquationSource Format="TEX">${{\mathbb {K}}}[x,x^{-1}]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">K</mi> <mo stretchy="false">[</mo> <mi>x</mi> <mo>,</mo> <msup> <mi>x</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>.</p>

On the centroid of a Leavitt path algebra

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2025年6月19日

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Daniel Gonçalves,
Dolores Martín Barquero,
Cándido Martín González,
Mercedes Siles Molina

摘要

We describe the centroid of some Leavitt path algebras. More precisely, for Leavitt path algebras over a field ${{\mathbb {K}}}$ $K$ , we show that if the algebra is simple, then its centroid is isomorphic to ${{\mathbb {K}}}$ $K$ , and if the algebra is prime, then its centroid is also isomorphic to ${{\mathbb {K}}}$ $K$ , except if the graph is a row-finite comet, in which case the centroid is isomorphic to ${{\mathbb {K}}}[x,x^{-1}]$ $K [x, x^{- 1}]$ .