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https:///mymemo.xyz/wiki/api.php?action=feedcontributions&feedformat=atom&user=114.49.7.26 2025-01-15T07:34:37Z 利用者の投稿記録 MediaWiki 1.31.0 2016-12-29T14:45:02Z

<p>114.49.7.26: </p> <hr /> <div>[[抽象代数学]]における&#039;&#039;&#039;十六元数&#039;&#039;&#039;（じゅうろくげんすう、&lt;em lang=&quot;en&quot;&gt;sedenion&lt;/em&gt;）は、全体として[[実数]]体 {{math|&#039;&#039;&#039;R&#039;&#039;&#039;}} 上{{math|16}}次元の（双線型な乗法を持つベクトル空間という意味での）非結合的[[多元環|分配多元環]]を成す代数的な対象で、その全体はしばしば {{math|&#039;&#039;&#039;S&#039;&#039;&#039;}} で表される。[[八元数]]に[[ケーリー＝ディクソンの構成法]]を使って得られる対合的二次代数である。<br /> <br /> 「十六元数」という用語は、他の十六次元代数構造、例えば[[四元数]]の複製二つの[[多元環のテンソル積|テンソル積]]や実数体上の四次正方行列環などに対しても用いられ、{{harvtxt|Smith |1995}} で調べられている。<br /> <br /> == 算術 ==<br /> ケーリーの八元数と同様に十六元数の乗法は[[交換法則|可換]]でも[[結合法則|結合的]]でもない。そして、ケーリーの八元数環 {{math|&#039;&#039;&#039;O&#039;&#039;&#039;}} と明確に違うことに、十六元数の全体 {{math|&#039;&#039;&#039;S&#039;&#039;&#039;}} は[[交代代数]]にもならない。十六元数についていえることは{{仮リンク|冪結合性|en|Power associativity}}を持っているということである。これは {{math|&#039;&#039;&#039;S&#039;&#039;&#039;}} の元 {{mvar|x}} に対して、冪 {{mvar|x{{exp|n}}}} は[[well-defined|矛盾なく定義可能]]で、それらが{{仮リンク|柔軟恒等式|label=柔軟|en|flexible identity}}であることを意味する。<br /> <br /> 任意の十六元数は、{{math|&#039;&#039;&#039;R&#039;&#039;&#039;}}-[[ベクトル空間]]としての {{math|&#039;&#039;&#039;S&#039;&#039;&#039;}} の[[基底 (線型代数学)|基底]]を成す16個の単位十六元数 {{math|{{mvar|e}}{{ind|0}} {{=}} 1, {{mvar|e}}{{ind|1}}, {{mvar|e}}{{ind|2}}, {{mvar|e}}{{ind|3}}, …, {{mvar|e}}{{ind|15}}}} の実係数[[線型結合]]になっている。<br /> <br /> 十六元数は乗法に関する単位元を持ち、多くの元がその逆元を持つが、[[多元体]]とはならない。これは[[零因子]]の存在による。つまり、それ自体は零ではないが掛けると零になるような十六元数の組があるのだが、簡単な例としては {{math|({{mvar|e}}{{ind|3}} + {{mvar|e}}{{ind|10}}) × ({{mvar|e}}{{ind|6}} − {{mvar|e}}{{ind|15}})}} などを挙げることができる。十六元数からケーリー＝ディクソンの構成法を元にして作られるどの超複素数系も零因子を含む。<br /> <br /> 単位十六元数の乗積表は次のようなものである。<br /> {| class=&quot;wikitable&quot; style=&quot;text-align:right; margin:1ex auto 2ex auto;&quot; align=center<br /> |+ style=&quot;font-weight:bold; font-size:larger;&quot; | 基底の乗積表<br /> |- <br /> ! {{math|&amp;times;}} !! {{math|1}} !! {{math|&#039;&#039;e&#039;&#039;{{ind|1}}}} !! {{math|&#039;&#039;e&#039;&#039;{{ind|2}}}} !! {{math|&#039;&#039;e&#039;&#039;{{ind|3}}}} !! {{math|&#039;&#039;e&#039;&#039;{{ind|4}}}} !! {{math|&#039;&#039;e&#039;&#039;{{ind|5}}}} !! {{math|&#039;&#039;e&#039;&#039;{{ind|6}}}} !! {{math|&#039;&#039;e&#039;&#039;{{ind|7}}}} !! {{math|&#039;&#039;e&#039;&#039;{{ind|8}}}} !! {{math|&#039;&#039;e&#039;&#039;{{ind|9}}}} !! {{math|&#039;&#039;e&#039;&#039;{{ind|10}}}} !! {{math|&#039;&#039;e&#039;&#039;{{ind|11}}}} !! {{math|&#039;&#039;e&#039;&#039;{{ind|12}}}} !! {{math|&#039;&#039;e&#039;&#039;{{ind|13}}}} !! {{math|&#039;&#039;e&#039;&#039;{{ind|14}}}} !! {{math|&#039;&#039;e&#039;&#039;{{ind|15}}}} <br /> |- <br /> ! {{math|1}} <br /> | {{math|1}} || {{math|&#039;&#039;e&#039;&#039;{{ind|1}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|2}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|3}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|4}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|5}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|6}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|7}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|8}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|9}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|10}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|11}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|12}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|13}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|14}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|15}}}}&lt;/td&gt;<br /> |- <br /> ! {{math|&#039;&#039;e&#039;&#039;{{ind|1}}}} <br /> | {{math|&#039;&#039;e&#039;&#039;{{ind|1}}}} || {{math|&amp;minus;1}} || {{math|&#039;&#039;e&#039;&#039;{{ind|3}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|2}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|5}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|4}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|7}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|6}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|9}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|8}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|11}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|10}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|13}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|12}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|15}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|14}}}}<br /> |- <br /> ! {{math|&#039;&#039;e&#039;&#039;{{ind|2}}}} <br /> | {{math|&#039;&#039;e&#039;&#039;{{ind|2}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|3}}}} || {{math|&amp;minus;1}} || {{math|&#039;&#039;e&#039;&#039;{{ind|1}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|6}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|7}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|4}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|5}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|10}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|11}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|8}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|9}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|14}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|15}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|12}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|13}}}}<br /> |- <br /> ! {{math|&#039;&#039;e&#039;&#039;{{ind|3}}}} <br /> | {{math|&#039;&#039;e&#039;&#039;{{ind|3}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|2}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|1}}}} || {{math|&amp;minus;1}} || {{math|&#039;&#039;e&#039;&#039;{{ind|7}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|6}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|5}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|4}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|11}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|10}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|9}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|8}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|15}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|14}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|13}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|12}}}}<br /> |- <br /> ! {{math|&#039;&#039;e&#039;&#039;{{ind|4}}}} <br /> | {{math|&#039;&#039;e&#039;&#039;{{ind|4}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|5}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|6}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|7}}}} || {{math|&amp;minus;1}} || {{math|&#039;&#039;e&#039;&#039;{{ind|1}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|2}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|3}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|12}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|13}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|14}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|15}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|8}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|9}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|10}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|11}}}}<br /> |- <br /> ! {{math|&#039;&#039;e&#039;&#039;{{ind|5}}}} <br /> | {{math|&#039;&#039;e&#039;&#039;{{ind|5}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|4}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|7}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|6}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|1}}}} || {{math|&amp;minus;1}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|3}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|2}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|13}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|12}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|15}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|14}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|9}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|8}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|11}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|10}}}}<br /> |- <br /> ! {{math|&#039;&#039;e&#039;&#039;{{ind|6}}}} <br /> | {{math|&#039;&#039;e&#039;&#039;{{ind|6}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|7}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|4}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|5}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|2}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|3}}}} || {{math|&amp;minus;1}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|1}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|14}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|15}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|12}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|13}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|10}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|11}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|8}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|9}}}}<br /> |- <br /> ! {{math|&#039;&#039;e&#039;&#039;{{ind|7}}}} <br /> | {{math|&#039;&#039;e&#039;&#039;{{ind|7}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|6}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|5}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|4}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|3}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|2}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|1}}}} || {{math|&amp;minus;1}} || {{math|&#039;&#039;e&#039;&#039;{{ind|15}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|14}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|13}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|12}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|11}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|10}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|9}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|8}}}}<br /> |- <br /> ! {{math|&#039;&#039;e&#039;&#039;{{ind|8}}}} <br /> | {{math|&#039;&#039;e&#039;&#039;{{ind|8}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|9}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|10}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|11}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|12}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|13}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|14}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|15}}}} || {{math|&amp;minus;1}} || {{math|&#039;&#039;e&#039;&#039;{{ind|1}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|2}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|3}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|4}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|5}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|6}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|7}}}}<br /> |- <br /> ! {{math|&#039;&#039;e&#039;&#039;{{ind|9}}}} <br /> | {{math|&#039;&#039;e&#039;&#039;{{ind|9}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|8}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|11}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|10}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|13}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|12}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|15}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|14}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|1}}}} || {{math|&amp;minus;1}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|3}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|2}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|5}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|4}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|7}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|6}}}}<br /> |- <br /> ! {{math|&#039;&#039;e&#039;&#039;{{ind|10}}}} <br /> | {{math|&#039;&#039;e&#039;&#039;{{ind|10}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|11}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|8}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|9}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|14}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|15}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|12}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|13}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|2}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|3}}}} || {{math|&amp;minus;1}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|1}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|6}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|7}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|4}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|5}}}}<br /> |- <br /> ! {{math|&#039;&#039;e&#039;&#039;{{ind|11}}}} <br /> | {{math|&#039;&#039;e&#039;&#039;{{ind|11}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|10}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|9}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|8}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|15}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|14}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|13}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|12}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|3}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|2}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|1}}}} || {{math|&amp;minus;1}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|7}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|6}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|5}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|4}}}}<br /> |- <br /> ! {{math|&#039;&#039;e&#039;&#039;{{ind|12}}}} <br /> | {{math|&#039;&#039;e&#039;&#039;{{ind|12}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|13}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|14}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|15}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|8}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|9}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|10}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|11}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|4}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|5}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|6}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|7}}}} || {{math|&amp;minus;1}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|1}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|2}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|3}}}}<br /> |- <br /> ! {{math|&#039;&#039;e&#039;&#039;{{ind|13}}}} <br /> | {{math|&#039;&#039;e&#039;&#039;{{ind|13}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|12}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|15}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|14}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|9}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|8}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|11}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|10}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|5}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|4}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|7}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|6}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|1}}}} || {{math|&amp;minus;1}} || {{math|&#039;&#039;e&#039;&#039;{{ind|3}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|2}}}}<br /> |- <br /> ! {{math|&#039;&#039;e&#039;&#039;{{ind|14}}}} <br /> | {{math|&#039;&#039;e&#039;&#039;{{ind|14}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|15}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|12}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|13}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|10}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|11}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|8}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|9}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|6}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|7}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|4}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|5}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|2}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|3}}}} || {{math|&amp;minus;1}} || {{math|&#039;&#039;e&#039;&#039;{{ind|1}}}}<br /> |- <br /> ! {{math|&#039;&#039;e&#039;&#039;{{ind|15}}}} <br /> | {{math|&#039;&#039;e&#039;&#039;{{ind|15}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|14}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|13}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|12}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|11}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|10}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|9}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|8}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|7}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|6}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|5}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|4}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|3}}}} || {{math|&#039;&#039;e&#039;&#039;{{ind|2}}}} || {{math|&amp;minus;&#039;&#039;e&#039;&#039;{{ind|1}}}} || &amp;minus;1<br /> |}<br /> <br /> 一般の十六元数の積は基底における乗法を（[[分配法則]]が成り立つように）線型に拡張することで得られる。<br /> <br /> 十六元数の全体 {{math|&#039;&#039;&#039;S&#039;&#039;&#039;}} は共軛元をとる主[[対合]]<br /> : &lt;math&gt;x = \sum_{i=0}^{15} x_i e_i \mapsto x^* := x_0 1 - \sum_{i=1}^{15} x_i e_i&lt;/math&gt;<br /> によってノルム<br /> : &lt;math&gt;N(x) := xx^* = \sum_{i=0}^{15} x_i^2\quad(\text{or }\|x\|:=\sqrt{xx^*})&lt;/math&gt;<br /> の定まる二次代数 {{math|(&#039;&#039;&#039;S&#039;&#039;&#039;, &#039;&#039;N&#039;&#039;)}} であるが、これは[[合成代数|ノルムが乗法的]]でない。<br /> <br /> == 応用 ==<br /> {{harvtxt|Moreno|1998}} はノルム 1 の十六元数からなる掛けて {{math|0}} となる対の全体が、例外型リー群 [[G2 (数学)|&#039;&#039;G&#039;&#039;&lt;sub&gt;2&lt;/sub&gt;]] のコンパクト型に同型であることを示した。<br /> <br /> == 関連項目 ==<br /> * [[超複素数系]]<br /> * [[分解型複素数]]<br /> <br /> == 参考文献 ==<br /> <br /> *{{Citation | last1=Imaeda | first1=K. | last2=Imaeda | first2=M. | title=Sedenions: algebra and analysis | doi=10.1016/S0096-3003(99)00140-X | mr=1786945 | year=2000 | journal=Applied mathematics and computation | volume=115 | issue=2 | pages=77–88}}<br /> * Kinyon, M.K., Phillips, J.D., Vojtěchovský, P.: &#039;&#039;C-loops: Extensions and constructions&#039;&#039;, Journal of Algebra and its Applications 6 (2007), no. 1, 1–20. [http://arxiv.org/abs/math/0412390]<br /> * Kivunge, Benard M. and Smith, Jonathan D. H: &quot;[http://www.emis.de/journals/CMUC/pdf/cmuc0402/kivunge.pdf Subloops of sedenions]&quot;, Comment.Math.Univ.Carolinae 45,2 (2004)295–302.<br /> *{{Citation | last1=Moreno | first1=Guillermo | title=The zero divisors of the Cayley-Dickson algebras over the real numbers | arxiv=q-alg/9710013 | mr=1625585 | year=1998 | journal=Sociedad Matemática Mexicana. Boletí n. Tercera Serie | volume=4 | issue=1 | pages=13–28}}<br /> *{{Citation | last1=Smith | first1=Jonathan D. H. | title=A left loop on the 15-sphere | doi=10.1006/jabr.1995.1237 | mr=1345298 | year=1995 | journal=[[Journal of Algebra]] | volume=176 | issue=1 | pages=128–138}}<br /> <br /> <br /> {{DEFAULTSORT:しゆうろくけんすう}}<br /> [[Category:超複素数系]]<br /> [[Category:非結合代数]]<br /> [[Category:数学に関する記事]]</div>

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