十六维类赛迪尼翁结合代数
Sixteen-dimensional Sedenion-like Associative Algebra
AI总结：
本文构造了作为$Cl_{5,0}$偶子代数的16维类赛迪尼翁结合代数，定义了其范数与对极等运算以构建Hopf代数结构，并证明了其范数保持性质及$\mathbb{Z}_2^5/2$-分级拟代数特性。
摘要：
本文构造了一个16维类赛迪尼翁结合代数，它是$2^5$维Clifford代数$Cl_{5,0}$的偶子代数。我们定义了类赛迪尼翁代数上的范数，并证明在条件$S_rS_d^\dagger + S_r^\dagger S_d=0$下，其16维元素保持范数关系$\lVert ST \rVert=\lVert S \rVert \lVert T \rVert$，其中$S_r,~S_d$分别表示八元数类数$S$的实部和对偶部，$S^\dagger$是$S$的转置。该类赛迪尼翁代数的元素可写为对偶八元数类数，称为分裂双八元数类代数，且对于任意两个八元数类数/赛迪尼翁类数$S$和$T$，$S S^\dagger$是可交换的[即$S S^\dagger=S^\dagger S $且$(S S^\dagger) T=T(S S^\dagger )$]。我们在八元数类/赛迪尼翁类代数上定义了余积$\bigtriangleup$、余单位$ε$和对极$S$运算，以在其上构造Hopf代数结构。我们还证明8维八元数类结合半范数除代数是$\mathbb{Z}_2^4/2$-分级的拟代数，16维类赛迪尼翁代数是$\mathbb{Z}_2^5/2$-分级的拟代数。
英文摘要：
In this article, we construct a $16$-dimensional sedenion-like associative algebra, which is an even subalgebra of $2^5$-dimensional Clifford algebra $Cl_{5,0}$. We define the norm on sedenion-like algebra and show that its sixteen-dimensional elements preserves the norm relation $\lVert ST \rVert=\lVert S \rVert \lVert T \rVert$ under the condition $S_rS_d^\dagger + S_r^\dagger S_d=0$, where $S_r,~S_d$ denote the real and dual part of an octonion-like number $S$ respectively and $S^\dagger$ is the transpose of $S$. The elements of this sedenion-like algebra can be written as dual octonion like numbers called split bioctonion-like algebra and $S S^\dagger$ is commutative [i.e. $S S^\dagger=S^\dagger S $ and $(S S^\dagger) T=T(S S^\dagger )$], for any two octonion-like/sedenion-like numbers $S$ and $T$. We define the operations coproduct $\bigtriangleup$, counit $ε$ and antipode $S$ on octonion-like/sedenion-like algebra to construct the Hopf algebra structure on it. We also show that $8$-dimensional octonion-like associative seminormed division algebra is a $\mathbb{Z}_2^4/2$-graded quasialgebra and $16$ dimensional sedenion-like algebra is a $\mathbb{Z}_2^5/2$-graded quasialgebra.
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