Affine Metric Geometry and Weak Orthogonal Groups
arxiv(2023)
摘要
By following the ideas underpinning the well-established ``homogeneous
model'' of an $n$-dimensional Euclidean space, we investigate whether the
motion group or the weak motion group of an $n$-dimensional affine metric space
on a vector space $V$ over an arbitrary field admits a specific faithful linear
representation as weak orthogonal group of an $(n+1)$-dimensional metric vector
space. Apart from a few exceptions, such a representation exists precisely when
the metric structure on $V$ is given by a quadratic form with a non-degenerate
polar form.
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