Sixteen-dimensional Sedenion-like Associative Algebra

Abstract

In this article, we construct a $16$-dimensional sedenion-like associative algebra, which is an even subalgebra of $2^5$-dimensional Clifford algebra $C{l}_{5,0}$. We define the norm on sedenion-like algebra and show that its sixteen-dimensional elements preserves the norm relation $\Vert ST\Vert =\Vert S\Vert \Vert T\Vert$ under the condition $S_r{S}_d^{†}+S_r^{†}{S}_d=0$, where $S_r,S_d$ denote the real and dual part of an octonion-like number $S$ respectively and $S^{†}$ 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 $S{S}^{†}$ is commutative [i.e. $S{S}^{†}=S^{†}S$ and $(S{S}^{†})T=T(S{S}^{†})$], for any two octonion-like/sedenion-like numbers $S$ and $T$. We define the operations coproduct $△$, counit $\epsilon$ 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.

15 pages