The Möbius Addition and Generalized Laplace–Beltrami Operator in Octonionic Space

The aim of this paper is to study the properties of the Möbius addition \(\oplus \) $\oplus$ under the action of the gyration operator gyr[a, b], and the relation between \((\sigma ,t)\) $(\sigma ,t)$ -translation defined by the Möbius addition and the generalized Laplace–Beltrami operator \(\Delta _{\sigma ,t} \) ${\mathrm{\Delta }}_{\sigma ,t}$ in the octonionic space. Despite the challenges posed by the non-associativity and non-commutativity of octonions, Möbius addition still exhibits many significant properties in the octonionic space, such as the left cancellation law and the gyrocommutative law. We introduce a novel approach to computing the Jacobian determinant of Möbius addition. Then, we discover that the gyration operator is closely related to the Jacobian matrix of Möbius addition. Importantly, we determine that the distinction between \(a\oplus x\) $a\oplus x$ and \(x\oplus a \) $x\oplus a$ is a specific orthogonal matrix factor. Finally, we demonstrate that the \((\sigma ,t)\) $(\sigma ,t)$ -translation is a unitary operator in \(L^2 \left( {\mathbb {B}^8_t,d\tau _{\sigma ,t} } \right) \) $L^2\left({\mathbb{B}}_t^8,d{\tau }_{\sigma ,t}\right)$ and it commutes with the generalized Laplace–Beltrami operator \(\Delta _{\sigma ,t} \) ${\mathrm{\Delta }}_{\sigma ,t}$ in the octonionic space.