MATHEMATICA BOHEMICA, Vol. 130, No. 3, pp. 265-275 (2005)

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

Barbara Glanc, Aleksander Misiak, Zofia Stepien

Barbara Glanc, Aleksander Misiak, Zofia Stepien, Instytut Matematyki, Politechnika Szczecinska, Al. Piastow 17, 70-310 Szczecin, e-mail:

Abstract: In this note all vectors and $\varepsilon$-vectors of a system of $m\leq n$ 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 $F( A{\underset1\to u}, A{\underset2\to u},\dots,A{\underset m\to u}) =( \det A)^{\lambda}\cdot A\cdot F( {\underset1\to u},{\underset2\to u},\dots, {\underset m\to u})$ with $\lambda=0$ and $\lambda=1$, for an arbitrary pseudo-orthogonal matrix $A$ of index one and given vectors $ {\underset1\to u},{\underset2\to u},\dots,{\underset m\to u}.$

Keywords: $G$-space, equivariant map, pseudo-Euclidean geometry

Classification (MSC2000): 53A55

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