ジャイロベクトル空間
1,On Central Extensions Of Gyrocommutative Gyrogroups
L.V. Sabinin (1995), "On the gyrogroups of Hungar", RUSS MATH SURV, 1995, 50 (5), 1095–1096.
L.V. Sabinin, L.L. Sabinina, Larissa Sbitneva (1998), Aequationes Mathematicae, On the notion of gyrogroup
L.V. Sabinin, Larissa Sbitneva, I.P. Shestakov (2006), "Non-associative Algebra and Its Applications",CRC Press,ISBN 0-8247-2669-3, 978-0-8247-2669-0
F. Smarandache, C. Barbu (2010), The Hyperbolic Menelaus Theorem in The Poincaré Disc Model of Hyperbolic Geometry
Roman Ulrich Sexl, Helmuth Kurt Urbantke, (2001), "Relativity, Groups, Particles: Special Relativity and Relativistic Symmetry in Field and Particle Physics", pages 141–142, Springer, ISBN 3-211-83443-5, 978-3-211-83443-5
ここで、 γ u = 1 1 − u 1 2 + u 2 2 + u 3 2 c 2 {\displaystyle \gamma _{\mathbf {u} }={\frac {1}{\sqrt {1-{\frac {u_{1}^{2}+u_{2}^{2}+u_{3}^{2}}{c^{2}}}}}}} である。
アインシュタイン加算は u {\displaystyle \mathbf {u} } と v {\displaystyle \mathbf {v} } が平行であるときのみ 可換かつ結合的である。実際、
u ⊕ v = g y r [ u , v ] ( v ⊕ u ) {\displaystyle \mathbf {u} \oplus \mathbf {v} =\mathrm {gyr} [\mathbf {u} ,\mathbf {v} ](\mathbf {v} \oplus \mathbf {u} )}