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Article

Glanc, Barbara ; Misiak, Aleksander ; Szmuksta-Zawadzka, Maria
Equivariant mappings from vector product into $ G$-spaces of $\varphi $-scalars with $G=O\left( n,1,\mathbb{R}\right) $. (English). Mathematica Bohemica, vol. 132 (2007), issue 3, pp. 325-332
MSC: 53A55 | MR 2355661 | Zbl 1174.53007 | DOI: 10.21136/MB.2007.134120
Keywords:
$G$-space; equivariant map; pseudo-Euclidean geometry
Summary:
There are four kinds of scalars in the $n$-dimensional pseudo-Euclidean geometry of index one. In this note, we determine all scalars as concomitants of a system of $m\le n$ linearly independent contravariant vectors of two so far missing types. The problem is resolved by finding the general solution of the functional equation $F( A\underset{1}{\rightarrow }{u},A \underset{2}{\rightarrow }{u},\dots ,A\underset{m}{\rightarrow }{u}) = \varphi \left( A\right) \cdot F( \underset{1}{\rightarrow }{u},\underset{2}{\rightarrow }{u},\dots ,\underset{m}{\rightarrow }{u})$ using two homomorphisms $\varphi $ from a group $G$ into the group of real numbers $\mathbb{R}_{0}=\left( \mathbb{R}\setminus \left\rbrace 0\right\lbrace ,\cdot \right)$.
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