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Measuring coalgebra
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<h2 id="definition">Definition</h2> <p>A coalgebra <i>C</i> with a linear map from <i>C</i>×<i>A</i> to <i>B</i> is said to measure <i>A</i> to <i>B</i> if it preserves the algebra product and identity (in the coalgebra sense). If we think of the elements of <i>C</i> as linear maps from <i>A</i> to <i>B</i>, this means that <i>c</i>(<i>a</i>1<i>a</i>2) = Σ<i>c</i>1(<i>a</i>1)<i>c</i>2(<i>a</i>2) where Σ<i>c</i>1⊗<i>c</i>2 is the coproduct of <i>c</i>, and <i>c</i> multiplies identities by the counit of <i>c</i>. In particular if <i>c</i> is grouplike this just states that <i>c</i> is a homomorphism from <i>A</i> to <i>B</i>. A measuring coalgebra is a universal coalgebra that measures <i>A</i> to <i>B</i> in the sense that any coalgebra that measures <i>A</i> to <i>B</i> can be mapped to it in a unique natural way. </p> <h2 id="examples">Examples</h2> <ul><li>The group-like elements of a measuring coalgebra from <i>A</i> to <i>B</i> are the homomorphisms from <i>A</i> to <i>B</i>.</li> <li>The primitive elements of a measuring coalgebra from <i>A</i> to <i>B</i> are the derivations from <i>A</i> to <i>B</i>.</li> <li>If <i>A</i> is the algebra of continuous real functions on a compact Hausdorff space <i>X</i>, and <i>B</i> is the real numbers, then the measuring coalgebra from <i>A</i> to <i>B</i> can be identified with finitely supported measures on <i>X</i>. This may be the origin of the term "measuring coalgebra".</li> <li>In the special case when <i>A</i> = <i>B</i>, the measuring coalgebra has a natural structure of a <a href="/facts/Hopf_algebra/3QF6g0tT">Hopf algebra</a>, called the Hopf algebra of the algebra <i>A</i>.</li></ul> <ul><li>Hazewinkel, Michiel; Gubareni, Nadiya; Kirichenko, V. V. (2010), <i>Algebras, rings and modules. Lie algebras and Hopf algebras</i>, Mathematical Surveys and Monographs, vol. 168, Providence, RI: American Mathematical Society, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8218-5262-0, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2724822">2724822</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1211.16023">1211.16023</a></li> <li>Sweedler, Moss E. (1968), "The Hopf algebra of an algebra applied to field theory", <i>J. Algebra</i>, 8 (3): 262–276, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2F0021-8693%2868%2990059-8">10.1016/0021-8693(68)90059-8</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0222053">0222053</a></li> <li>Sweedler, Moss E. (1969), <a href="https://books.google.com/books?id=8FnvAAAAMAAJ"><i>Hopf algebras</i></a>, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9780805392548, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0252485">0252485</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0194.32901">0194.32901</a></li></ul>