<?xml version="1.0" encoding="ISO-8859-1"?><article xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">
<front>
<journal-meta>
<journal-id>0120-419X</journal-id>
<journal-title><![CDATA[Revista Integración]]></journal-title>
<abbrev-journal-title><![CDATA[Integración - UIS]]></abbrev-journal-title>
<issn>0120-419X</issn>
<publisher>
<publisher-name><![CDATA[Universidad Industrial de Santander]]></publisher-name>
</publisher>
</journal-meta>
<article-meta>
<article-id>S0120-419X2021000100007</article-id>
<article-id pub-id-type="doi">10.18273/revint.v39n1-2021007</article-id>
<title-group>
<article-title xml:lang="en"><![CDATA[Some special types of determinants in graded skew PBW extensions]]></article-title>
<article-title xml:lang="es"><![CDATA[Algunos tipos especiales de determinantes en extensiones PBW torcidas graduadas]]></article-title>
</title-group>
<contrib-group>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Suárez]]></surname>
<given-names><![CDATA[Héctor]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
</contrib>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Cáceres]]></surname>
<given-names><![CDATA[Duban]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
</contrib>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Reyes]]></surname>
<given-names><![CDATA[Armando]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
</contrib>
</contrib-group>
<aff id="Af1">
<institution><![CDATA[,Universidad Pedagógica y Tecnológica de Colombia Escuela de Matemáticas y Estadística ]]></institution>
<addr-line><![CDATA[ ]]></addr-line>
<country>Colombia</country>
</aff>
<aff id="Af2">
<institution><![CDATA[,Universidad Pedagógica y Tecnológica de Colombia Escuela de Matemáticas y Estadística ]]></institution>
<addr-line><![CDATA[ ]]></addr-line>
<country>Colombia</country>
</aff>
<aff id="Af3">
<institution><![CDATA[,Universidad Nacional de Colombia Departamento de Matemáticas ]]></institution>
<addr-line><![CDATA[ ]]></addr-line>
<country>Colombia</country>
</aff>
<pub-date pub-type="pub">
<day>00</day>
<month>06</month>
<year>2021</year>
</pub-date>
<pub-date pub-type="epub">
<day>00</day>
<month>06</month>
<year>2021</year>
</pub-date>
<volume>39</volume>
<numero>1</numero>
<fpage>91</fpage>
<lpage>107</lpage>
<copyright-statement/>
<copyright-year/>
<self-uri xlink:href="http://www.scielo.org.co/scielo.php?script=sci_arttext&amp;pid=S0120-419X2021000100007&amp;lng=en&amp;nrm=iso"></self-uri><self-uri xlink:href="http://www.scielo.org.co/scielo.php?script=sci_abstract&amp;pid=S0120-419X2021000100007&amp;lng=en&amp;nrm=iso"></self-uri><self-uri xlink:href="http://www.scielo.org.co/scielo.php?script=sci_pdf&amp;pid=S0120-419X2021000100007&amp;lng=en&amp;nrm=iso"></self-uri><abstract abstract-type="short" xml:lang="en"><p><![CDATA[Abstract In this paper, we prove that the Nakayama automorphism of a graded skew PBW extension over a finitely presented Koszul Auslanderregular algebra has trivial homological determinant. For A = &#963;(R) {x1, x2} a graded skew PBW extension over a connected algebra R, we compute its Pdeterminant and the inverse of &#963;. In the particular case of quasi-commutative skew PBW extensions over Koszul Artin-Schelter regular algebras, we show explicitly the connection between the Nakayama automorphism of the ring of coefficients and the extension. Finally, we give conditions to guarantee that A is Calabi-Yau. We provide illustrative examples of the theory concerning algebras of interest in noncommutative algebraic geometry and noncommutative differential geometry.]]></p></abstract>
<abstract abstract-type="short" xml:lang="es"><p><![CDATA[Resumen En este artículo, demostramos que el automorfismo de Nakayama de una extensión PBW torcida graduada sobre un álgebra de Koszul finitamente presentada y Auslander-regular tiene determinante homológico trivial. Para A = &#963;(R) {x1, x2} una extensión PBW torcida graduada sobre un álgebra conexa R, calculamos su P-determinante y el inverso de &#963;. En el caso particular de extensiones PBW torcidas cuasi-conmutativas sobre álgebras de Koszul Artin-Schelter regulares, mostramos explícitamente la relación entre el automorfismo de Nakayama del anillo de coeficientes y la extensión. Finalmente, damos condiciones para garantizar que A sea Calabi-Yau. Proporcionamos ejemplos ilustrativos de la teoría con álgebras de interés en geometría algebraica no conmutativa y geometría diferencial no conmutativa.]]></p></abstract>
<kwd-group>
<kwd lng="en"><![CDATA[Calabi-Yau algebra]]></kwd>
<kwd lng="en"><![CDATA[skew PBW extension]]></kwd>
<kwd lng="en"><![CDATA[double Ore extension]]></kwd>
<kwd lng="en"><![CDATA[homological determinant]]></kwd>
<kwd lng="en"><![CDATA[P-determinant]]></kwd>
<kwd lng="en"><![CDATA[Nakayama automorphism]]></kwd>
<kwd lng="en"><![CDATA[MSC2010: 16S37]]></kwd>
<kwd lng="en"><![CDATA[16W50]]></kwd>
<kwd lng="en"><![CDATA[16W70]]></kwd>
<kwd lng="en"><![CDATA[16S36]]></kwd>
<kwd lng="en"><![CDATA[13N10]]></kwd>
<kwd lng="es"><![CDATA[Álgebra Calabi-Yau]]></kwd>
<kwd lng="es"><![CDATA[extensión PBW torcida]]></kwd>
<kwd lng="es"><![CDATA[extensión de Ore doble]]></kwd>
<kwd lng="es"><![CDATA[determinante homológico]]></kwd>
<kwd lng="es"><![CDATA[P-determinante]]></kwd>
<kwd lng="es"><![CDATA[automorfismo de Nakayama]]></kwd>
</kwd-group>
</article-meta>
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