八元数空间中的Mbius加法与旋转算子

The Mobius addition and gyration operator in the octonionic space

Mbius加法⊕与旋转算子gyr[a,b]在旋转群理论以及双曲几何中发挥着重要的作用.本文将Mbius加法和旋转算子推广到了八元数空间中.虽然八元数乘法是非交换非结合的,但旋转算子却能在一定程度上修复Mbius加法缺失的交换性,产生了旋转交换律.不仅如此,本文还为八元数空间上的Mbius加法赋予了其他丰富的内容,如左循环性和左消去律等.而由Mbius加法和旋转算子导出的Mbius协同加法在八元数空间中满足更一般的交换律,即,a■b=b■a.

Mobius addition⊕and gyration operator gyr[a,b]play an important role in gyrogroup theory and hyperbolic geometry.In this paper,the Mobius addition and gyration operator are extended to the octonionic space.Although the multiplication of octonions is non-commutative and non-associative,the gyration operator can repair the missing commutativity of Mobius addition,resulting in the gyrocommutative law.In addition,the gyration operator gives rich content to Mobius addition on the octonion,for example,the left loop property and the left cancellation law.The Mobius coaddition derived from the Mobius addition and gyration operator can satisfy the general commutative law in the octonionic space,i.e.,a⊕b=b⊕a.