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Rotations in the space of Split octonions. (English) Zbl 1201.81066
Summary: The geometrical application of split octonions is considered. The new representation of products of the basis units of split octonionic having David’s star shape (instead of the Fano triangle) is presented. It is shown that active and passive transformations of coordinates in octonionic “eight-space” are not equivalent. The group of passive transformations that leave invariant the pseudonorm of split octonions is SO$$(4,4)$$, while active rotations are done by the direct product of $$O(3,4)$$-boosts and real noncompact form of the exceptional group $$G_{2}$$. In classical limit, these transformations reduce to the standard Lorentz group.
##### MSC:
 81R05 Finite-dimensional groups and algebras motivated by physics and their representations 83A05 Special relativity 11R52 Quaternion and other division algebras: arithmetic, zeta functions 70H40 Relativistic dynamics for problems in Hamiltonian and Lagrangian mechanics 17B25 Exceptional (super)algebras 17A35 Nonassociative division algebras
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