十六元數

在抽象代数中，十六元數（英語：Sedenion）是在實數上形成的16維非交換且非結合代數結構。彷如八元數，其乘法不符合交換律及結合律。十六元數可以透過將八元數套用凯莱-迪克森结构來構造。然而，与八元数不一样，十六元数甚至不符合交错性。尽管如此，十六元数仍然符合幂结合性。此外，十六元數中存在零因子（zero divisor），例如${\displaystyle {{\left({{\,e_{3}}+{\,e_{10}}}\right)}\times {\left({{\,e_{6}}-{\,e_{15}}}\right)}}=0}$，這點與八元數截然不同——因此，十六元數無法構成整環（integral domain），也無法構成除環（divisor ring）。^[1]

十六元數
符號	${\displaystyle \mathbb {S} }$
種類	非结合代數
單位	${\displaystyle e_{0}}$、 ${\displaystyle e_{1}}$、 ${\displaystyle e_{2}}$、 ${\displaystyle e_{3}}$、 ${\displaystyle e_{4}}$、 ${\displaystyle e_{5}}$、 ${\displaystyle e_{6}}$、 ${\displaystyle e_{7}}$、 ${\displaystyle e_{8}}$、 ${\displaystyle e_{9}}$、 ${\displaystyle e_{10}}$、 ${\displaystyle e_{11}}$、 ${\displaystyle e_{12}}$、 ${\displaystyle e_{13}}$、 ${\displaystyle e_{14}}$ 及${\displaystyle e_{15}}$
乘法單位元	${\displaystyle e_{0}}$
主要性質	冪結合性 分配律
常見的數字系統
${\displaystyle \mathbb {N} }$ 自然數 ${\displaystyle \mathbb {Z} }$ 整數 ${\displaystyle \mathbb {Q} }$ 有理數 ${\displaystyle \mathbb {R} }$ 實數 ${\displaystyle \mathbb {C} }$ 複數 ${\displaystyle \mathbb {H} }$ 四元數 少見的數字系統 八元數 (${\displaystyle \mathbb {O} }$) 十六元數 (${\displaystyle \mathbb {S} }$) 三十二元數
查 论 编

各种各样的数

十六元數是由八元數套用凯莱-迪克森構造而成的。十六元數亦可以繼續進行凯莱-迪克森構造。若將十六元數套用凯莱-迪克森構造將會形成三十二元數（trigintaduonion）。^[2]每一次的構造都會導致維數翻倍^[3]:45，並且構造結果同樣與十六元數類似，有著不符合交错性、符合幂结合性與存在零因子等特性。^[3]

十六元數這個術語同時亦用於其他同為16維度的代數結構，例如兩個複四元數的張量積、實數上的4×4矩陣代數或喬納森·D·H·史密斯於1995提出的一種代數結構。^[4]

算術

 

立方八元數擴展到四維空間的視覺化^[5]:6，其展示了35個示例十六元數之實數${\displaystyle (e_{0})}$ 頂點三元組所構成的超平面。唯一的例外是三元組${\displaystyle (e_{1})}$ , ${\displaystyle (e_{2})}$ , ${\displaystyle (e_{3})}$ 不與${\displaystyle (e_{0})}$ 形成超平面。

十六元數的乘法和八元數一樣，不具備交換律及結合律。與八元數不同的是，十六元數不具備交错代数的特性。雖然如此，但十六元數仍然保有冪結合性，也就是說，對所有的十六元數集${\displaystyle \mathbb {S} }$ 中的元素x，冪${\displaystyle x^{n}}$ 是可以明確定義的。同時，十六元數亦有柔性代數（英语：Flexible algebra）的特性。^[6]

十六元數共有的16個單位。這16個單位十六元數是：^[7]

${\displaystyle e_{0}}$ 、${\displaystyle e_{1}}$ 、${\displaystyle e_{2}}$ 、${\displaystyle e_{3}}$ 、${\displaystyle e_{4}}$ 、${\displaystyle e_{5}}$ 、${\displaystyle e_{6}}$ 、${\displaystyle e_{7}}$ 、${\displaystyle e_{8}}$ 、${\displaystyle e_{9}}$ 、${\displaystyle e_{10}}$ 、${\displaystyle e_{11}}$ 、${\displaystyle e_{12}}$ 、${\displaystyle e_{13}}$ 、${\displaystyle e_{14}}$ 及${\displaystyle e_{15}}$ 

每個十六元數都是單位十六元數${\displaystyle e_{0}}$ , ${\displaystyle e_{1}}$ , ${\displaystyle e_{2}}$ , ${\displaystyle e_{3}}$ , ..., ${\displaystyle e_{15}}$ 的線性組合，並構成了十六元數向量空間的基。 每個十六元數都可以用以下形式表示：^[7]

${\displaystyle x=x_{0}e_{0}+x_{1}e_{1}+x_{2}e_{2}+\cdots +x_{14}e_{14}+x_{15}e_{15}.}$ 

十六元數的加法和減法是通過將相應十六元數單位之係數的加法或減法來定義的。而十六元數的乘法是對加法的分配，所以兩個十六元數的乘積可以通過對所有項的乘積求和來計算。^[1]

十六元數和其他也由凯莱-迪克森结构來構造的代數結構一樣，其皆包含了依凯莱-迪克森结构構造來源的代數結構。例如十六元數可透過八元數代凯莱-迪克森结构來構造、八元數可透過四元數代凯莱-迪克森结构來構造、四元數可透過複數代凯莱-迪克森结构來構造、複數可透過實數代凯莱-迪克森结构來構造。因此，十六元數系包含了一個八元數系（由下方乘法表對應的${\displaystyle e_{0}}$ 至${\displaystyle e_{7}}$ 構造），亦包含了四元數系（由${\displaystyle e_{0}}$ 至${\displaystyle e_{3}}$ 構造），也包含了複數系（由${\displaystyle e_{0}}$ 至${\displaystyle e_{1}}$ 構造）和實數系（由${\displaystyle e_{0}}$ 構造）。^[8]

十六元數具有乘法單位元素${\displaystyle e_{0}}$ 和乘法逆元，但因為存在零因子因此無法構成可除代數（英语：Division algebra）。換句話說，即十六元數的代數系統中，存在2個非零十六元數相乘為零，例如${\displaystyle (e_{3}+e_{10})(e_{6}-e_{15})=0}$ 。其他基於凯莱-迪克森结构構造的超複數系統中，維度大於16的超複數也都存在零因子。^[7][1]

十六元數單元乘數表如下：^[8]

${\displaystyle e_{i}e_{j}}$		${\displaystyle e_{j}}$
		${\displaystyle e_{0}}$	${\displaystyle e_{1}}$	${\displaystyle e_{2}}$	${\displaystyle e_{3}}$	${\displaystyle e_{4}}$	${\displaystyle e_{5}}$	${\displaystyle e_{6}}$	${\displaystyle e_{7}}$	${\displaystyle e_{8}}$	${\displaystyle e_{9}}$	${\displaystyle e_{10}}$	${\displaystyle e_{11}}$	${\displaystyle e_{12}}$	${\displaystyle e_{13}}$	${\displaystyle e_{14}}$	${\displaystyle e_{15}}$
${\displaystyle e_{i}}$	${\displaystyle e_{0}}$	${\displaystyle e_{0}}$	${\displaystyle e_{1}}$	${\displaystyle e_{2}}$	${\displaystyle e_{3}}$	${\displaystyle e_{4}}$	${\displaystyle e_{5}}$	${\displaystyle e_{6}}$	${\displaystyle e_{7}}$	${\displaystyle e_{8}}$	${\displaystyle e_{9}}$	${\displaystyle e_{10}}$	${\displaystyle e_{11}}$	${\displaystyle e_{12}}$	${\displaystyle e_{13}}$	${\displaystyle e_{14}}$	${\displaystyle e_{15}}$
	${\displaystyle e_{1}}$	${\displaystyle e_{1}}$	${\displaystyle -e_{0}}$	${\displaystyle e_{3}}$	${\displaystyle -e_{2}}$	${\displaystyle e_{5}}$	${\displaystyle -e_{4}}$	${\displaystyle -e_{7}}$	${\displaystyle e_{6}}$	${\displaystyle e_{9}}$	${\displaystyle -e_{8}}$	${\displaystyle -e_{11}}$	${\displaystyle e_{10}}$	${\displaystyle -e_{13}}$	${\displaystyle e_{12}}$	${\displaystyle e_{15}}$	${\displaystyle -e_{14}}$
	${\displaystyle e_{2}}$	${\displaystyle e_{2}}$	${\displaystyle -e_{3}}$	${\displaystyle -e_{0}}$	${\displaystyle e_{1}}$	${\displaystyle e_{6}}$	${\displaystyle e_{7}}$	${\displaystyle -e_{4}}$	${\displaystyle -e_{5}}$	${\displaystyle e_{10}}$	${\displaystyle e_{11}}$	${\displaystyle -e_{8}}$	${\displaystyle -e_{9}}$	${\displaystyle -e_{14}}$	${\displaystyle -e_{15}}$	${\displaystyle e_{12}}$	${\displaystyle e_{13}}$
	${\displaystyle e_{3}}$	${\displaystyle e_{3}}$	${\displaystyle e_{2}}$	${\displaystyle -e_{1}}$	${\displaystyle -e_{0}}$	${\displaystyle e_{7}}$	${\displaystyle -e_{6}}$	${\displaystyle e_{5}}$	${\displaystyle -e_{4}}$	${\displaystyle e_{11}}$	${\displaystyle -e_{10}}$	${\displaystyle e_{9}}$	${\displaystyle -e_{8}}$	${\displaystyle -e_{15}}$	${\displaystyle e_{14}}$	${\displaystyle -e_{13}}$	${\displaystyle e_{12}}$
	${\displaystyle e_{4}}$	${\displaystyle e_{4}}$	${\displaystyle -e_{5}}$	${\displaystyle -e_{6}}$	${\displaystyle -e_{7}}$	${\displaystyle -e_{0}}$	${\displaystyle e_{1}}$	${\displaystyle e_{2}}$	${\displaystyle e_{3}}$	${\displaystyle e_{12}}$	${\displaystyle e_{13}}$	${\displaystyle e_{14}}$	${\displaystyle e_{15}}$	${\displaystyle -e_{8}}$	${\displaystyle -e_{9}}$	${\displaystyle -e_{10}}$	${\displaystyle -e_{11}}$
	${\displaystyle e_{5}}$	${\displaystyle e_{5}}$	${\displaystyle e_{4}}$	${\displaystyle -e_{7}}$	${\displaystyle e_{6}}$	${\displaystyle -e_{1}}$	${\displaystyle -e_{0}}$	${\displaystyle -e_{3}}$	${\displaystyle e_{2}}$	${\displaystyle e_{13}}$	${\displaystyle -e_{12}}$	${\displaystyle e_{15}}$	${\displaystyle -e_{14}}$	${\displaystyle e_{9}}$	${\displaystyle -e_{8}}$	${\displaystyle e_{11}}$	${\displaystyle -e_{10}}$
	${\displaystyle e_{6}}$	${\displaystyle e_{6}}$	${\displaystyle e_{7}}$	${\displaystyle e_{4}}$	${\displaystyle -e_{5}}$	${\displaystyle -e_{2}}$	${\displaystyle e_{3}}$	${\displaystyle -e_{0}}$	${\displaystyle -e_{1}}$	${\displaystyle e_{14}}$	${\displaystyle -e_{15}}$	${\displaystyle -e_{12}}$	${\displaystyle e_{13}}$	${\displaystyle e_{10}}$	${\displaystyle -e_{11}}$	${\displaystyle -e_{8}}$	${\displaystyle e_{9}}$
	${\displaystyle e_{7}}$	${\displaystyle e_{7}}$	${\displaystyle -e_{6}}$	${\displaystyle e_{5}}$	${\displaystyle e_{4}}$	${\displaystyle -e_{3}}$	${\displaystyle -e_{2}}$	${\displaystyle e_{1}}$	${\displaystyle -e_{0}}$	${\displaystyle e_{15}}$	${\displaystyle e_{14}}$	${\displaystyle -e_{13}}$	${\displaystyle -e_{12}}$	${\displaystyle e_{11}}$	${\displaystyle e_{10}}$	${\displaystyle -e_{9}}$	${\displaystyle -e_{8}}$
	${\displaystyle e_{8}}$	${\displaystyle e_{8}}$	${\displaystyle -e_{9}}$	${\displaystyle -e_{10}}$	${\displaystyle -e_{11}}$	${\displaystyle -e_{12}}$	${\displaystyle -e_{13}}$	${\displaystyle -e_{14}}$	${\displaystyle -e_{15}}$	${\displaystyle -e_{0}}$	${\displaystyle e_{1}}$	${\displaystyle e_{2}}$	${\displaystyle e_{3}}$	${\displaystyle e_{4}}$	${\displaystyle e_{5}}$	${\displaystyle e_{6}}$	${\displaystyle e_{7}}$
	${\displaystyle e_{9}}$	${\displaystyle e_{9}}$	${\displaystyle e_{8}}$	${\displaystyle -e_{11}}$	${\displaystyle e_{10}}$	${\displaystyle -e_{13}}$	${\displaystyle e_{12}}$	${\displaystyle e_{15}}$	${\displaystyle -e_{14}}$	${\displaystyle -e_{1}}$	${\displaystyle -e_{0}}$	${\displaystyle -e_{3}}$	${\displaystyle e_{2}}$	${\displaystyle -e_{5}}$	${\displaystyle e_{4}}$	${\displaystyle e_{7}}$	${\displaystyle -e_{6}}$
	${\displaystyle e_{10}}$	${\displaystyle e_{10}}$	${\displaystyle e_{11}}$	${\displaystyle e_{8}}$	${\displaystyle -e_{9}}$	${\displaystyle -e_{14}}$	${\displaystyle -e_{15}}$	${\displaystyle e_{12}}$	${\displaystyle e_{13}}$	${\displaystyle -e_{2}}$	${\displaystyle e_{3}}$	${\displaystyle -e_{0}}$	${\displaystyle -e_{1}}$	${\displaystyle -e_{6}}$	${\displaystyle -e_{7}}$	${\displaystyle e_{4}}$	${\displaystyle e_{5}}$
	${\displaystyle e_{11}}$	${\displaystyle e_{11}}$	${\displaystyle -e_{10}}$	${\displaystyle e_{9}}$	${\displaystyle e_{8}}$	${\displaystyle -e_{15}}$	${\displaystyle e_{14}}$	${\displaystyle -e_{13}}$	${\displaystyle e_{12}}$	${\displaystyle -e_{3}}$	${\displaystyle -e_{2}}$	${\displaystyle e_{1}}$	${\displaystyle -e_{0}}$	${\displaystyle -e_{7}}$	${\displaystyle e_{6}}$	${\displaystyle -e_{5}}$	${\displaystyle e_{4}}$
	${\displaystyle e_{12}}$	${\displaystyle e_{12}}$	${\displaystyle e_{13}}$	${\displaystyle e_{14}}$	${\displaystyle e_{15}}$	${\displaystyle e_{8}}$	${\displaystyle -e_{9}}$	${\displaystyle -e_{10}}$	${\displaystyle -e_{11}}$	${\displaystyle -e_{4}}$	${\displaystyle e_{5}}$	${\displaystyle e_{6}}$	${\displaystyle e_{7}}$	${\displaystyle -e_{0}}$	${\displaystyle -e_{1}}$	${\displaystyle -e_{2}}$	${\displaystyle -e_{3}}$
	${\displaystyle e_{13}}$	${\displaystyle e_{13}}$	${\displaystyle -e_{12}}$	${\displaystyle e_{15}}$	${\displaystyle -e_{14}}$	${\displaystyle e_{9}}$	${\displaystyle e_{8}}$	${\displaystyle e_{11}}$	${\displaystyle -e_{10}}$	${\displaystyle -e_{5}}$	${\displaystyle -e_{4}}$	${\displaystyle e_{7}}$	${\displaystyle -e_{6}}$	${\displaystyle e_{1}}$	${\displaystyle -e_{0}}$	${\displaystyle e_{3}}$	${\displaystyle -e_{2}}$
	${\displaystyle e_{14}}$	${\displaystyle e_{14}}$	${\displaystyle -e_{15}}$	${\displaystyle -e_{12}}$	${\displaystyle e_{13}}$	${\displaystyle e_{10}}$	${\displaystyle -e_{11}}$	${\displaystyle e_{8}}$	${\displaystyle e_{9}}$	${\displaystyle -e_{6}}$	${\displaystyle -e_{7}}$	${\displaystyle -e_{4}}$	${\displaystyle e_{5}}$	${\displaystyle e_{2}}$	${\displaystyle -e_{3}}$	${\displaystyle -e_{0}}$	${\displaystyle e_{1}}$
	${\displaystyle e_{15}}$	${\displaystyle e_{15}}$	${\displaystyle e_{14}}$	${\displaystyle -e_{13}}$	${\displaystyle -e_{12}}$	${\displaystyle e_{11}}$	${\displaystyle e_{10}}$	${\displaystyle -e_{9}}$	${\displaystyle e_{8}}$	${\displaystyle -e_{7}}$	${\displaystyle e_{6}}$	${\displaystyle -e_{5}}$	${\displaystyle -e_{4}}$	${\displaystyle e_{3}}$	${\displaystyle e_{2}}$	${\displaystyle -e_{1}}$	${\displaystyle -e_{0}}$

十六元數特性

從上表可得到：

對所有的${\displaystyle i}$ ，有${\displaystyle e_{0}e_{i}=e_{i}e_{0}=e_{i}}$ ，

${\displaystyle e_{i}e_{i}=-e_{0}\,\,{\text{for}}\,\,i\neq 0}$ ，且

${\displaystyle e_{i}e_{j}=-e_{j}e_{i}\,\,{\text{for}}\,\,i\neq j\,\,{\text{with}}\,\,i,j\neq 0}$ 。

反結合

十六元數並非完全反結合。選擇任意四個生成元${\displaystyle i,j,k}$ 和${\displaystyle l}$ ，對於乘積${\displaystyle ijkl}$ ，有五種添加括號的方法。假如反結合律總是成立，則五者之間應有以下關係：

${\displaystyle (ij)(kl)=-((ij)k)l=(i(jk))l=-i((jk)l)=i(j(kl))=-(ij)(kl),}$ 

從而${\displaystyle (ij)(kl)=0}$ ，矛盾。所以，某兩者之間不滿足反結合律。

特別地，代入${\displaystyle e_{1},e_{2},e_{4}}$ 和${\displaystyle e_{8}}$ 時，利用上列乘法表，可得最後兩式滿足結合律：${\displaystyle e_{1}(e_{2}e_{12})=(e_{1}e_{2})e_{12}=-e_{15}}$ 。

四元子代數

在下表列出了構成這個特定十六元數乘法表的35個三元組。用於使用凯莱-迪克森结构構造之十六元數的7個八元數三元組，以粗體表示：

每個三元組中，三個數的二進制表示，按位異或的結果為0。

{ {1, 2, 3}, {1, 4, 5}, {1, 7, 6}, {1, 8, 9}, {1, 11, 10}, {1, 13, 12}, {1, 14, 15},
{2, 4, 6}, {2, 5, 7}, {2, 8, 10}, {2, 9, 11}, {2, 14, 12}, {2, 15, 13}, {3, 4, 7},
{3, 6, 5}, {3, 8, 11}, {3, 10, 9}, {3, 13, 14}, {3, 15, 12}, {4, 8, 12}, {4, 9, 13},
{4, 10, 14}, {4, 11, 15}, {5, 8, 13}, {5, 10, 15}, {5, 12, 9}, {5, 14, 11}, {6, 8, 14},
{6, 11, 13}, {6, 12, 10}, {6, 15, 9}, {7, 8, 15}, {7, 9, 14}, {7, 12, 11}, {7, 13, 10} }

84組由十六元數單位組成的零因子數組${\displaystyle \{e_{a},e_{b},e_{c},e_{d}\}}$ 列舉如下，其中 ${\displaystyle (e_{a}+e_{b})\circ (e_{c}+e_{d})=0}$ ：

一月 05, 2023