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

Document Type: Special Issue: International Conference on Architecture and Mathematics

Authors

Department of Mathematics, North Dakota State University, Fargo, ND 58105, USA

Abstract

‎The decomposition $\Gamma=BH$ of a group $\Gamma$ into a subset B ‎and a subgroup $H$ of $\Gamma$ 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 ‎\mbox{operation} given by the Einstein velocity addition law of ‎special relativity theory‎. ‎The latter leads to the Lorentz ‎transformation group $\so{1,n}$‎, ‎$n\in\N$‎, ‎in pseudo-Euclidean ‎spaces of signature $(1‎, ‎n)$‎. ‎The study in this article is motivated ‎by generalized Lorentz groups $\so{m‎, ‎n}$‎, ‎$m‎, ‎n\in\N$‎, ‎in ‎pseudo-Euclidean spaces of signature $(m‎, ‎n)$‎. ‎Accordingly‎, ‎this ‎article explores the bi-decomposition $\Gamma = H_LBH_R$ of a group ‎$\Gamma$ into a subset $B$ and subgroups $H_L$ and $H_R$ of ‎$\Gamma$‎, ‎along with the novel bi-gyrogroup structure of $B$ induced ‎by the bi-decomposition of $\Gamma$‎. ‎As an example‎, ‎we show by ‎methods of Clifford \mbox{algebras} that the quotient group of the ‎spin group $\spin{m‎, ‎n}$ possesses the bi-decomposition structure‎.

Keywords


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