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Micron Document

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<title>Koalgebra</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Koalgebra</h1>

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Eine Koalgebra ist ein <a href="Vektorraum" title="Vektorraum">Vektorraum</a>, der die zu einer Algebra duale Struktur besitzt. Das heißt anstelle einer Multiplikation, die zwei Elemente auf ihr Produkt abbildet, gibt es eine Komultiplikation, die ein Element auf ein <a href="Tensorprodukt" title="Tensorprodukt">Tensorprodukt</a> abbildet, und anstelle eines neutralen Elements, das die Einbettung des Grundkörpers in die Algebra ermöglicht, gibt es eine Abbildung aus der Koalgebra in den Grundkörper, die Koeins genannt wird.

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

Eine Koalgebra über einem <a href="K%C3%B6rper_(Algebra)" title="Körper (Algebra)">Körper</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> ist ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">-<a href="Vektorraum" title="Vektorraum">Vektorraum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

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<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> mit <a href="Vektorraumhomomorphismus" class="mw-redirect" title="Vektorraumhomomorphismus">Vektorraumhomomorphismen</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{C}\colon C\to C\otimes _{k}C}">

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<annotation encoding="application/x-tex">{\displaystyle \Delta _{C}\colon C\to C\otimes _{k}C}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f34ac4b7fff408913eaa94fc67d5694e75b52938.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.293ex; height:2.509ex;" alt="{\displaystyle \Delta _{C}\colon C\to C\otimes _{k}C}" loading="lazy">, genannt Komultiplikation, Koprodukt oder auch Diagonale, und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon _{C}\colon C\to k}">

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<annotation encoding="application/x-tex">{\displaystyle \epsilon _{C}\colon C\to k}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fdacf635ee109f57a70074ebd757c3585087bace.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.051ex; height:2.509ex;" alt="{\displaystyle \epsilon _{C}\colon C\to k}" loading="lazy">, genannt Koeins, so dass

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<annotation encoding="application/x-tex">{\displaystyle (\mathrm {id} _{C}\otimes \Delta _{C})\circ \Delta _{C}=(\Delta _{C}\otimes \mathrm {id} _{C})\circ \Delta _{C}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a11ed6d26aad44094f20280858bd0fad42a6a408.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.297ex; height:2.843ex;" alt="{\displaystyle (\mathrm {id} _{C}\otimes \Delta _{C})\circ \Delta _{C}=(\Delta _{C}\otimes \mathrm {id} _{C})\circ \Delta _{C}}" loading="lazy"> (Koassoziativität)</dd>

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<annotation encoding="application/x-tex">{\displaystyle (\mathrm {id} _{C}\otimes \epsilon _{C})\circ \Delta _{C}=\mathrm {id} _{C}=(\epsilon _{C}\otimes \mathrm {id} _{C})\circ \Delta _{C}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4a363e7740e5dbc5e65bf5d5f60186d6a48891f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.833ex; height:2.843ex;" alt="{\displaystyle (\mathrm {id} _{C}\otimes \epsilon _{C})\circ \Delta _{C}=\mathrm {id} _{C}=(\epsilon _{C}\otimes \mathrm {id} _{C})\circ \Delta _{C}}" loading="lazy"> (Koeins)</dd></dl>

Ein Koalgebrahomomorphismus zwischen zwei Koalgebren C und D ist ein Vektorraumhomomorphismus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon C\to D}">

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<annotation encoding="application/x-tex">{\displaystyle f\colon C\to D}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/40483a186caa819b1272ee13a7b258b57608cfff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.617ex; height:2.509ex;" alt="{\displaystyle f\colon C\to D}" loading="lazy"> mit

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<annotation encoding="application/x-tex">{\displaystyle f\otimes f\circ \Delta _{C}=\Delta _{D}\circ f}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6348613febf0bb21d4474cf41695bc49649aa915.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.11ex; height:2.509ex;" alt="{\displaystyle f\otimes f\circ \Delta _{C}=\Delta _{D}\circ f}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon _{C}=\epsilon _{D}\circ f}">

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<annotation encoding="application/x-tex">{\displaystyle \epsilon _{C}=\epsilon _{D}\circ f}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/13b70df2a3db3999222ee7bc3e6004b74b625152.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.534ex; height:2.509ex;" alt="{\displaystyle \epsilon _{C}=\epsilon _{D}\circ f}" loading="lazy">.</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Beispiel">Beispiel</h2></div>

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (e_{1},e_{2},e_{3})}">

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<annotation encoding="application/x-tex">{\displaystyle (e_{1},e_{2},e_{3})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1d9174b8943b9b8379276d0cf9b61fbbf21c0d59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.29ex; height:2.843ex;" alt="{\displaystyle (e_{1},e_{2},e_{3})}" loading="lazy"> die kanonische Basis von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f936ddf584f8f3dd2a0ed08917001b7a404c10b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}" loading="lazy">. Man kann auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}">

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<annotation encoding="application/x-tex">{\displaystyle \Delta _{\mathbb {R} ^{3}}(e_{i})=e_{i}\otimes e_{i}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/795383d49b49cf54f25f59709740198824291b96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.584ex; height:3.009ex;" alt="{\displaystyle \Delta _{\mathbb {R} ^{3}}(e_{i})=e_{i}\otimes e_{i}}" loading="lazy"></dd></dl>

und

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<annotation encoding="application/x-tex">{\displaystyle \epsilon _{\mathbb {R} ^{3}}(e_{i})=1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2ddb459fbd177f6c2ee92c04cb83243d3b5e09a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.148ex; height:3.009ex;" alt="{\displaystyle \epsilon _{\mathbb {R} ^{3}}(e_{i})=1}" loading="lazy"></dd></dl>

definieren.

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<annotation encoding="application/x-tex">{\displaystyle \Delta _{\mathbb {R} ^{3}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7f40548e14fb538ef33856c60e260381006eb349.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.186ex; height:2.843ex;" alt="{\displaystyle \Delta _{\mathbb {R} ^{3}}}" loading="lazy"> ist koassoziativ, da

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<annotation encoding="application/x-tex">{\displaystyle e_{i}\otimes \Delta _{\mathbb {R} ^{3}}(e_{i})=e_{i}\otimes e_{i}\otimes e_{i}=\Delta _{\mathbb {R} ^{3}}(e_{i})\otimes e_{i}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/387e247dc91f0e17f123e417b1fd9b88c38efdb6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:42.732ex; height:3.009ex;" alt="{\displaystyle e_{i}\otimes \Delta _{\mathbb {R} ^{3}}(e_{i})=e_{i}\otimes e_{i}\otimes e_{i}=\Delta _{\mathbb {R} ^{3}}(e_{i})\otimes e_{i}}" loading="lazy">,</dd></dl>

und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon _{\mathbb {R} ^{3}}}">

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<annotation encoding="application/x-tex">{\displaystyle \epsilon _{\mathbb {R} ^{3}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e9a26468c8ebc66615006cc9567d88fef8c48317.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.195ex; height:2.343ex;" alt="{\displaystyle \epsilon _{\mathbb {R} ^{3}}}" loading="lazy"> ist Koeins, da

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<annotation encoding="application/x-tex">{\displaystyle e_{i}\otimes \epsilon _{\mathbb {R} ^{3}}(e_{i})=e_{i}=\epsilon _{\mathbb {R} ^{3}}(e_{i})\otimes e_{i}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/afbb982dc66920cd6c0d9c909e2ad856fe658df3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.301ex; height:3.009ex;" alt="{\displaystyle e_{i}\otimes \epsilon _{\mathbb {R} ^{3}}(e_{i})=e_{i}=\epsilon _{\mathbb {R} ^{3}}(e_{i})\otimes e_{i}}" loading="lazy">.</dd></dl>

Die Elemente von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}\otimes \mathbb {R} ^{3}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}\otimes \mathbb {R} ^{3}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0512ae6e9438c4931d378b4947c8981d7b3ddb0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.305ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} ^{3}\otimes \mathbb {R} ^{3}}" loading="lazy"> sind Tensoren zweiter Stufe und können daher als Matrizen dargestellt werden. Die Komultiplikation ist dann

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<annotation encoding="application/x-tex">{\displaystyle \Delta _{\mathbb {R} ^{3}}{\begin{pmatrix}a_{1}\\a_{2}\\a_{3}\end{pmatrix}}=a_{1}\Delta _{\mathbb {R} ^{3}}{\begin{pmatrix}1\\0\\0\end{pmatrix}}+a_{2}\Delta _{\mathbb {R} ^{3}}{\begin{pmatrix}0\\1\\0\end{pmatrix}}+a_{3}\Delta _{\mathbb {R} ^{3}}{\begin{pmatrix}0\\0\\1\end{pmatrix}}={\begin{pmatrix}a_{1}&0&0\\0&a_{2}&0\\0&0&a_{3}\end{pmatrix}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3b9be3a0d14c0c5087dc09addd11144c9b840791.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:76.835ex; height:9.176ex;" alt="{\displaystyle \Delta _{\mathbb {R} ^{3}}{\begin{pmatrix}a_{1}\\a_{2}\\a_{3}\end{pmatrix}}=a_{1}\Delta _{\mathbb {R} ^{3}}{\begin{pmatrix}1\\0\\0\end{pmatrix}}+a_{2}\Delta _{\mathbb {R} ^{3}}{\begin{pmatrix}0\\1\\0\end{pmatrix}}+a_{3}\Delta _{\mathbb {R} ^{3}}{\begin{pmatrix}0\\0\\1\end{pmatrix}}={\begin{pmatrix}a_{1}&0&0\\0&a_{2}&0\\0&0&a_{3}\end{pmatrix}}}" loading="lazy">.</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Dualität">Dualität</h2></div>

Die Multiplikation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{A}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a5f50df49aa96673fa1a3853443075a6ac869499.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.867ex; height:2.176ex;" alt="{\displaystyle \mu _{A}}" loading="lazy"> einer (unitären assoziativen) <a href="Algebra_(Struktur)" class="mw-redirect" title="Algebra (Struktur)">Algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> ist bilinear, und aufgrund der <a href="Universelle_Eigenschaft" title="Universelle Eigenschaft">Universellen Eigenschaft</a> des Tensorprodukts kann sie als Abbildung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\otimes A}">

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<annotation encoding="application/x-tex">{\displaystyle A\otimes A}</annotation>

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Eine Algebra <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> besitzt genau dann ein neutrales Element, wenn es einen Vektorraumhomomorphismus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta _{A}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \eta _{A}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f8674d5bf48359b4f2e0d67b55bcf19c720864f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.62ex; height:2.176ex;" alt="{\displaystyle \eta _{A}}" loading="lazy"> gibt, so dass das folgende Diagramm kommutiert:

In diesem Fall gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1_{A}=\eta _{A}(1_{k})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1_{A}=\eta _{A}(1_{k})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/79fa442f4fcff8cedf27a191a423e6f530ae95ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.407ex; height:2.843ex;" alt="{\displaystyle 1_{A}=\eta _{A}(1_{k})}" loading="lazy">.

Eine Koalgebra <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> ist eine Algebra in der zu den Vektorräumen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Vekt} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {Vekt} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/00f9e6efc8cf97b83a5575bccef1b62fb49111b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.908ex; height:2.176ex;" alt="{\displaystyle \mathrm {Vekt} }" loading="lazy"> <a href="Duale_Kategorie" title="Duale Kategorie">dualen Kategorie</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Vekt} ^{\mathrm {op} }}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {Vekt} ^{\mathrm {op} }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/44c15b484735e3af863858b231585f61c8b4542c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.876ex; height:2.343ex;" alt="{\displaystyle \mathrm {Vekt} ^{\mathrm {op} }}" loading="lazy">. Das heißt, anstelle der Multiplikation gibt es eine Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{C}\colon C\to C\otimes C}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta _{C}\colon C\to C\otimes C}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b057997679cb6ed9967e0b33f51d9c4d3b040622.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.205ex; height:2.509ex;" alt="{\displaystyle \Delta _{C}\colon C\to C\otimes C}" loading="lazy">, so dass das folgende duale Diagramm kommutiert:

Und anstelle eines neutralen Elements gibt es eine Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon _{C}\colon C\to k}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \epsilon _{C}\colon C\to k}</annotation>

</semantics>

<div class="mw-heading mw-heading2"><h2 id="Sweedlernotation">Sweedlernotation</h2></div>

Über das Koprodukt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{C}(x)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta _{C}(x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c2263c8b6b31e2d808ca76874fec3919790ec4af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.556ex; height:2.843ex;" alt="{\displaystyle \Delta _{C}(x)}" loading="lazy"> eines Elements <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in C}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f7fc788379bff7289bdf694ffd68ee690e999eb7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.937ex; height:2.176ex;" alt="{\displaystyle x\in C}" loading="lazy"> ist im Allgemeinen nur bekannt, dass es in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C\otimes C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C\otimes C}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5168f0f4d33b5dc3b4c5ac1dc8150439a312acc0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.373ex; height:2.343ex;" alt="{\displaystyle C\otimes C}" loading="lazy"> liegt und sich folglich als

<msub>

</mrow>

</msub>

</mrow>

</munder>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta _{C}(x)=\sum _{i}x_{(1)}^{(i)}\otimes x_{(2)}^{(i)}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1edc27fe2117b49c70672c47040b9fd7387af26e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:23.564ex; height:5.676ex;" alt="{\displaystyle \Delta _{C}(x)=\sum _{i}x_{(1)}^{(i)}\otimes x_{(2)}^{(i)}}" loading="lazy"></dd></dl>

darstellen lässt. In der Sweedler-Notation (nach <a href="Moss_Sweedler" title="Moss Sweedler">Moss Sweedler</a>) wird dies abgekürzt, indem man symbolisch

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta _{C}(x)=\sum _{(x)}x_{(1)}\otimes x_{(2)}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b27741685e992c9c99a791ae50c0cf94ac77eeb9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:23.564ex; height:6.009ex;" alt="{\displaystyle \Delta _{C}(x)=\sum _{(x)}x_{(1)}\otimes x_{(2)}}" loading="lazy"></dd></dl>

schreibt. In summenloser Sweedler-Notation verzichtet man sogar auf das Summensymbol und schreibt

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta _{C}(x)=x_{(1)}\otimes x_{(2)}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f64881fa50347b377935b61102ead641b51b637a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:19.822ex; height:3.176ex;" alt="{\displaystyle \Delta _{C}(x)=x_{(1)}\otimes x_{(2)}}" loading="lazy"></dd></dl>

Es ist dabei wichtig zu beachten, dass diese Schreibweise nach wie vor eine Summe bezeichnet. Die Symbole <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{(1)}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{(1)}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/464c1aa643240b7e29d9a21be0f4c1c75d1c20d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.663ex; height:2.509ex;" alt="{\displaystyle x_{(1)}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{(2)}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{(2)}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/80fefeb7856adb89bb51dd37d20691ef3c57d791.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.663ex; height:2.509ex;" alt="{\displaystyle x_{(2)}}" loading="lazy"> sind für sich allein bedeutungslos und stehen nicht für bestimmte Elemente aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy">, denn die Darstellung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \Delta _{C}(x)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \Delta _{C}(x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/50e707a023bf09e3b9f2fdb1da5f4a952f64999e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.556ex; height:2.843ex;" alt="{\displaystyle \textstyle \Delta _{C}(x)}" loading="lazy"> ist nicht eindeutig. Bei Rechnungen in der Sweedlernotation liest man die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle x_{(k)}^{(i)}}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle x_{(k)}^{(i)}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d67fe2e12b68374b4f5d334ebea1dc2b24acc9bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:3.698ex; height:4.176ex;" alt="{\displaystyle \textstyle x_{(k)}^{(i)}}" loading="lazy"> am besten als „geeignete und für diese Rechnung fest gewählte“ Elemente.

Diese Schreibweise ermöglicht es, die Komposition von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{C}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta _{C}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b21ef96f46482e08181ef4ef5d3d83c290977a5b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.417ex; height:2.509ex;" alt="{\displaystyle \Delta _{C}}" loading="lazy"> mit anderen Funktionen als

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (f\otimes g)\circ \Delta _{C}(x)=f(x_{(1)})\otimes g(x_{(2)})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2d911a2a065624b9edb0f582ab6ac1d5153380f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:35.074ex; height:3.176ex;" alt="{\displaystyle (f\otimes g)\circ \Delta _{C}(x)=f(x_{(1)})\otimes g(x_{(2)})}" loading="lazy"></dd></dl>

zu schreiben.

In summenloser Sweedler-Notation ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon _{C}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \epsilon _{C}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b7bf5921e796d0c6f5f9134bd9c3d43365bf1610.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.425ex; height:2.009ex;" alt="{\displaystyle \epsilon _{C}}" loading="lazy"> genau dann Koeins, wenn

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \epsilon _{C}(x_{(1)})x_{(2)}=x=x_{(1)}\epsilon _{C}(x_{(2)})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5fe60b7b71e27201863692c8fe533b82dde9b5dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:30.649ex; height:3.176ex;" alt="{\displaystyle \epsilon _{C}(x_{(1)})x_{(2)}=x=x_{(1)}\epsilon _{C}(x_{(2)})}" loading="lazy">.</dd></dl>

Das Koprodukt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{C}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta _{C}}</annotation>

</semantics>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{(1)}\otimes \Delta _{C}(x_{(2)})=\Delta _{C}(x_{(1)})\otimes x_{(2)}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/61cc7cc1b53deaa0799b79cc5671779495606a88.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:33.885ex; height:3.176ex;" alt="{\displaystyle x_{(1)}\otimes \Delta _{C}(x_{(2)})=\Delta _{C}(x_{(1)})\otimes x_{(2)}}" loading="lazy">.</dd></dl>

Dieses Element wird in Sweedler-Notation symbolisch als

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{(x)}x_{(1)}\otimes x_{(2)}\otimes x_{(3)}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bdfada1d59f00683fee59ecea52490ede55b8592.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:20.413ex; height:6.009ex;" alt="{\displaystyle \sum _{(x)}x_{(1)}\otimes x_{(2)}\otimes x_{(3)}}" loading="lazy"></dd></dl>

und summenlos als

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{(1)}\otimes x_{(2)}\otimes x_{(3)}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/89a68ad62d9fdc5fb7280e2f10e7898a48c97851.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:16.671ex; height:2.843ex;" alt="{\displaystyle x_{(1)}\otimes x_{(2)}\otimes x_{(3)}}" loading="lazy"></dd></dl>

geschrieben.

Durch erneutes Anwenden von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{C}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta _{C}}</annotation>

</semantics>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x_{(1)})\otimes \Delta _{C}(x_{(2)})\otimes g(x_{(3)})=f(x_{(1)})\otimes x_{(2)}\otimes x_{(3)}\otimes g(x_{(4)})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/443ead6618ef98ff2561cff07f9cdfdb2c2a2f92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:60.196ex; height:3.176ex;" alt="{\displaystyle f(x_{(1)})\otimes \Delta _{C}(x_{(2)})\otimes g(x_{(3)})=f(x_{(1)})\otimes x_{(2)}\otimes x_{(3)}\otimes g(x_{(4)})}" loading="lazy">.</dd></dl>

Durch Anwenden von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon _{C}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \epsilon _{C}}</annotation>

</semantics>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x_{(1)})\otimes \epsilon _{C}(x_{(2)})\otimes x_{(3)}\otimes g(x_{(4)})=f(x_{(1)})\otimes x_{(2)}\otimes g(x_{(3)})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/130b6b0e58ccc3a58fa6d564a5cc578e81033dec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:59.205ex; height:3.176ex;" alt="{\displaystyle f(x_{(1)})\otimes \epsilon _{C}(x_{(2)})\otimes x_{(3)}\otimes g(x_{(4)})=f(x_{(1)})\otimes x_{(2)}\otimes g(x_{(3)})}" loading="lazy">.</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>

<ul><li>Christian Kassel: Quantum Groups In: Graduate Texts in Mathematics. Springer-Verlag, ISBN 0-387-94370-6.</li></ul></div><div><div class="zim-footer">

Dieser Artikel wurde von <a class="external text" title="Zuletzt bearbeitet am 2024-06-17" href="https://de.wikipedia.org/wiki/?title=Koalgebra&oldid=245999791">Wikipedia</a> herausgegeben. Der Text ist unter <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.de">Creative Commons Attribution-Share Alike 4.0</a> verfügbar, sofern nicht anders angegeben. Für die Mediendateien können zusätzliche Bedingungen gelten.

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