By Zhukavets N.M., Shestakov I.P.
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Additional info for A base of the free alternative superalgebra on one odd generator
As there are only three lines, it is quite natural to try to connect the vectors with the special basis. This basis is unique up to permutation and the form (2) given above defines the value of an arbitrary number module and also the length of the vector, – all being of the simplest shape. Concerning the originality of such basis, we will give it a proper name of the Absolute basis. In this respect the concerned space turns out to be arranged in an absolutely another way, than the usual Euclidean and pseudo-Euclidean spaces, where there are no preferred bases (with an exception of the pseudo-Euclidean plane), and that is why we usually try to turn the studying of analogous geometries into a non-coordinate form.
Secondly, there are several concepts of the total product of the plane vectors, and the majority of them have the inverse ones; meanwhile in other pseudo-Euclidean spaces only scalar product is introduced, as well as division is not defined at all. Thirdly, isotropic vectors always divide the pseudo-Euclidean planes with the signature (1, n − 1) into 3 simply connected domains, with an exception of the plane, with 4 such domains. Fourthly, it does not matter which of the two typical coordinates of the Euclidean space we will choose as the temporal and which as the spatial, as the result will change to permutation.
It is surprising that among all the Euclidean spaces only the two-dimensional is distinguished with its unique peculiarities, it is worth mentioning the following. Firstly, the theorem of Liouville, that enumerates the types of possible conformal transformations, coming to translations, rotations, dilatations and inversions, is true for all the pseudo-Euclidean spaces with 3 or more dimensions. In the two-dimensional case the list of their conformal transformations is by far longer. Secondly, there are several concepts of the total product of the plane vectors, and the majority of them have the inverse ones; meanwhile in other pseudo-Euclidean spaces only scalar product is introduced, as well as division is not defined at all.