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A homogeneous description of inhomogeneous Minkowski groups. (English) Zbl 0558.51011
The motion group of an n-dimensional metric geometry is $${\mathfrak M}(A)=0(A)\cdot {\mathfrak T}(A)$$, where A is an n-dimensional regular vector space, 0(A) the group of motions fixing the origin, and $${\mathfrak T}(A)$$ is the translation group. This is the familiar description of isometries by inhomogeneous coordinates. For many purposes a description of the same group by homogeneous coordinates or as a subgroup of the general linear group is more convenient.
In order to obtain the desired description we consider the space $$V=A\oplus R$$, where R is the radical of V and dim R$$=1$$. Let $$0^*(V)$$ be the subgroup of isometries whose elements fix every vector in R, $$0^+(V)$$ the subgroup of 0(V) whose elements have determinant 1 and $$Z=\{1_ V,-1_ V\}.$$ Then we obtain the following results: $$0^*(V)\cong {\mathfrak M}(A),0(V)/Z\cong {\mathfrak S}(A),$$ where $${\mathfrak S}(A)$$ is the group of similarities of A, $$0^+(V)\cong {\mathfrak M}(A)$$ if dim V is odd and $$0^+(V)/Z\cong {\mathfrak M}^+(V)$$ if dim V is even.
##### MSC:
 51F25 Orthogonal and unitary groups in metric geometry 51N25 Analytic geometry with other transformation groups 20H15 Other geometric groups, including crystallographic groups
##### Keywords:
Minkowski space; Euclidean space; similarities; orthogonal group
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