Einstein Gyrogroup as a B-loop

Abstract

Using the Clifford algebra formalism, we give an algebraic proof that the open unit ball B = { v ∈ R n : ‖ v ‖ 1 } of R n equipped with Einstein addition ⊕E forms a B-loop or, equivalently, a uniquely 2-divisible gyrocommutative gyrogroup. We obtain a compact formula for Einstein addition in terms of Mobius addition. We then give a characterization of associativity and commutativity of vectors in B with respect to Einstein addition.