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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
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