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            <title xml:lang="en">The alternating presentation of $U_q(\widehat{gl_2})$ from Freidel-Maillet algebras</title>
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            <idno type="halRefHtml">&lt;i&gt;Nuclear Physics B&lt;/i&gt;, 2021, 967 (June 2021), pp.115400. &lt;a target="_blank" href="https://dx.doi.org/10.1016/j.nuclphysb.2021.115400"&gt;&amp;#x27E8;10.1016/j.nuclphysb.2021.115400&amp;#x27E9;&lt;/a&gt;</idno>
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            <idno type="stamp" n="INSMI">CNRS-INSMI - INstitut des Sciences Mathématiques et de leurs Interactions</idno>
            <idno type="stamp" n="IDP">Institut Denis Poisson</idno>
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                <title xml:lang="en">The alternating presentation of $U_q(\widehat{gl_2})$ from Freidel-Maillet algebras</title>
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                <title level="j">Nuclear Physics B</title>
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                  <publisher>North-Holland ; Elsevier [1967-....]</publisher>
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                <term xml:lang="en">q-shuffle algebra</term>
                <term xml:lang="en">$U_q(\widehat{gl_2})$</term>
                <term xml:lang="en">Drinfeld second presentation</term>
                <term xml:lang="en">Reflection equation</term>
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              <p>An infinite dimensional algebra denoted $\bar{\cal A}_q$ that is isomorphic to a central extension of $U_q^+$ - the positive part of $U_q(\widehat{sl_2})$ - has been recently proposed by Paul Terwilliger. It provides an `alternating' Poincar\'e-Birkhoff-Witt (PBW) basis besides the known Damiani's PBW basis built from positive root vectors. In this paper, a presentation of $\bar{\cal A}_q$ in terms of a Freidel-Maillet algebra is obtained. Using this presentation: (a) finite dimensional tensor product representations for $\bar{\cal A}_q$ are constructed; (b) explicit isomorphisms from $\bar{\cal A}_q$ to certain Drinfeld type `alternating' subalgebras of $U_q(\widehat{gl_2})$ are obtained; (c) the image in $U_q^+$ of all the generators of $\bar{\cal A}_q$ in terms of Damiani's root vectors is obtained. A new tensor product decomposition of $U_q(\widehat{sl_2})$ in terms of Drinfeld type `alternating' subalgebras follows. The specialization $q\rightarrow 1$ of $\bar{\cal A}_q$ is also introduced and studied in details. In this case, a presentation is given as a non-standard Yang-Baxter algebra. This paper is dedicated to Paul Terwilliger for his 65th birthday.</p>
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