Bi-Gyrogroup: The Group-Like Structure Induced by Bi-Decomposition of Groups

Abstract

‎The decomposition Γ=BH of a group Γ into a subset B ‎and a subgroup H of Γ induces‎, ‎under general conditions‎, ‎a ‎group-like structure for B‎, ‎known as a gyrogroup‎. ‎The famous‎ concrete realization of a gyrogroup‎, ‎which motivated the emergence ‎of gyrogroups into the mainstream‎, ‎is the space of all ‎relativistically admissible velocities along with a binary ‎operation given by the Einstein velocity addition law of ‎special relativity theory‎. ‎The latter leads to the Lorentz ‎transformation group So(1,n)‎, ‎n∈N‎, ‎in pseudo-Euclidean ‎spaces of signature (1‎, ‎n)‎. ‎The study in this article is motivated ‎by generalized Lorentz groups So(m‎, ‎n)‎, ‎m‎, ‎n∈N, ‎in ‎pseudo-Euclidean spaces of signature (m‎, ‎n)‎. ‎Accordingly‎, ‎this ‎article explores the bi-decomposition Γ= HLBHR of a group ‎Γ into a subset B and subgroups HL and HR of ‎Γ‎, ‎along with the novel bi-gyrogroup structure of B induced ‎by the bi-decomposition of Γ‎. ‎As an example‎, ‎we show by ‎methods of Clifford algebras that the quotient group of the ‎spin group Spin(m‎, ‎n) possesses the bi-decomposition structure‎.

Keywords

• bi-decomposition of group‎
• ‎bi-gyrogroup‎
• ‎gyrogroup‎
• spin group‎
• ‎pseudo-orthogonal group‎