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Crystallographic concepts for inhomogeneous subgroups of the Poincaré group

Publication Date
DOI: 10.1016/0031-8914(71)90064-4
  • Mathematics


Abstract Concepts typical for crystallographic space groups, like the group of primitive translations ( U), the point group ( K) and the system of non-primitive translations ( u ), are also very convenient in the more general case of arbitrary inhomogeneous subgroups G of the Poincaré group IO(3, 1). Groups with given U and K are considered. The rôle of the cohomology groups H 1( K, R 4/ U) -where R 4 is the group of all space-time translations - and H 1( K, U) is discussed. H 1( K, R 4/ U) appears if one considers the imbedding of G into IO(3, 1), whereas H 2( K, U) occurs if one looks at G as extension of U by K. Not every such extension gives, in general, a subgroup of IO(3, 1). The elements of H 1( K, R 4/ U) are in one-to-one correspondence with the classes of subgroups G having given U and K, and only differing in their origins. If H 1( K, R 4) = 0 and if U generates the real vector space R 4, Bieberbach's conjugation theorem holds, i.e. abstract isomorphisms can be realized as conjugations in the affine group A(4). The important consequences of this property are considered and a number of basic theorems proved. Several physical systems having inhomogeneous subgroups of the Poincaré group as symmetry groups are indicated and discussed.

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