Equivariant mappings from vector product into $G$-space of vectors and $\varepsilon $-vectors with $G=O(n,1,\mathbb{R})$

Glanc, Barbara ; Misiak, Aleksander ; Stepień, Zofia

Mathematica Bohemica, Tome 130 (2005) no. 3, pp. 265-275.

Résumé

In this note all vectors and $\varepsilon $-vectors of a system of $m\le n$ linearly independent contravariant vectors in the $n$-dimensional pseudo-Euclidean geometry of index one are determined. The problem is resolved by finding the general solution of the functional equation $F( A{\underset{1}{\rightarrow }u}, A{\underset{2}{\rightarrow }u},\dots ,A{\underset{m}{\rightarrow }u}) =( \det A)^{\lambda }\cdot A\cdot F( {\underset{1}{\rightarrow }u},{\underset{2}{\rightarrow }u},\dots , {\underset{m}{\rightarrow }u})$ with $\lambda =0$ and $\lambda =1$, for an arbitrary pseudo-orthogonal matrix $A$ of index one and given vectors $ {\underset{1}{\rightarrow }u},{\underset{2}{\rightarrow }u},\dots ,{\underset{m}{\rightarrow }u}.$

Détail

DOI : 10.21136/MB.2005.134097

Classification : 22E99, 53A35, 53A55
Keywords: $G$-space; equivariant map; pseudo-Euclidean geometry; functional equation