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            <title xml:lang="en">The doubles of a braided Hopf algebra</title>
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                <forename type="first">Alain</forename>
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            <idno type="halRefHtml">&lt;i&gt;Contemporary mathematics&lt;/i&gt;, 2013, 585, pp.175-198. &lt;a target="_blank" href="https://dx.doi.org/10.1090/conm/585"&gt;&amp;#x27E8;10.1090/conm/585&amp;#x27E9;&lt;/a&gt;</idno>
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            <idno type="stamp" n="IMAG-MONTPELLIER">Institut Montpelliérain Alexander Grothendieck</idno>
            <idno type="stamp" n="UNIV-MONTPELLIER">Université de Montpellier</idno>
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                <title xml:lang="en">The doubles of a braided Hopf algebra</title>
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                    <forename type="first">Alain</forename>
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                <title level="j">Contemporary mathematics</title>
                <title level="m">Hopf Algebras and Tensor Categories</title>
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                  <title>Hopf Algebras and Tensor Categories</title>
                  <date type="start">2011-07-04</date>
                  <date type="end">2011-07-08</date>
                  <settlement>Almeria</settlement>
                  <country key="ES">Spain</country>
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              <p>Let A be a Hopf algebra in a braided rigid category B. In the case B admits a coend C, which is a Hopf algebra in B, we defined in 2008 the double D(A) of A, which is a quasitriangular Hopf algebra in B whose category of modules is isomorphic to the center of the category of A-modules as a braided category. Here, quasitriangular means endowed with an R-matrix (our notion of R-matrix for a Hopf algebra in B involves the coend C of B). In general, i.e. when B does not necessarily admit a coend, we construct a quasitriangular Hopf monad d_A on the center Z(B) of B whose category of modules is isomorphic to the center of the category of A-modules as a braided category. We prove that the Hopf monad d_A may not be representable by a Hopf algebra. If B has a coend C, then D(A) is the cross product of the Hopf monad d_A by C. Equivalently, the Hopf monad d_A is the cross quotient of the Hopf algebra D(A) by the Hopf algebra C.</p>
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