Retraction article: Eight-dimensional octonion-like but associative normed division algebra

Abstract

We present an eight-dimensional even sub-algebra of the $2^4=16$-dimensional associative Clifford algebra $C{l}_{4,0}$ and show that its eight-dimensional elements denoted as $\mathbf{X}$ and $\mathbf{Y}$ respect the norm relation $\Vert \mathbf{XY}\Vert =\Vert \mathbf{X}\Vert \Vert \mathbf{Y}\Vert ,$ thus forming an octonion-like but associative normed division algebra, where the norms are calculated using the fundamental geometric product instead of the usual scalar product. The corresponding 7-sphere has a topology that differs from that of octonionic 7-sphere.

Keywords:

• Associative algebras
• Clifford algebras
• Geometric Algebra
• norm division algebras
• octonions

Mathematics Subject Classifications:

• 11R52
• 17C60

View correction statement:

Statement of retraction: Eight-dimensional octonion-like but associative normed division algebra

Notes

1 The conformal space we are considering is an in-homogeneous version of the space usually studied in Conformal Geometric Algebra [5]. It can be viewed as an 8-dimensional subspace of the 32-dimensional representation space postulated in Conformal Geometric Algebra. The larger representation space results from a homogeneous freedom of the origin within ${\mathbb{E}}^3,$ which does not concern us in this article.