Semidirect Products

The fact that px is a semidirect product of these two subalgebras is a necessary condition to support such an interpretation. Indeed, since we have [p,p] = p, we see that the role played by the generators of symmetries p is to impress dynamical modification on the observables p giving rise to other observables. As a consequence, the non-commutativity between the observables is a matter of measurement. In the case we are studying in this section we have [p,p] = p resulting in a quantum theory. For the sake of consistency, we expect to derive a classical TFD theory with an algebra similar to pr but in which [p,p] = 0. (This result has been explored in Ref.(L.M. Silva et.al., 1997))... 

Note that in semidirect products the invariant subgroup is always the first group in the product. For example,... 

Evaluate the following DPs showing the symmetry operators in each group. [Hint For (a)-(e), evaluate products using projection diagrams. This technique is not useful for products that involve operators associated with the C3 axes of a cube or tetrahedron, so in these cases study the transformations induced in a cube.] Explain why the DPs in (d)-(f) are semidirect products. 

In this case the point subgroup (A10) of G is identical with the point group P, and G may be written as the semidirect product... 

In non-symmorphic space groups the point group P is not a subgroup of G and it is not possible to express G as a semidirect product. 

From Eq. (2.82) the semidirect product structure of any other of the representations of rW listed in Fig. 1 may be derived by the first isomorphy theorem22. For example for the representation... 

The semidirect product of the two groups (2.90) and (2.92) may be considered as the analogue of the permutation-inversion groups1 for molecules with non-trivial covering groups ( )... 

In this chapter we investigate various types of products arising naturally in scheme theory. We define direct products of closed subsets of S, direct products of schemes, quasi-direct products of schemes, and semidirect products of schemes. 

We call S A the semidirect product of S and A with respect to (. The scheme S is called the kernel of the semidirect product SfA, A its complement. 

Occasionally, we do not need to specify the group homomorphism ( which defines the semidirect product A. In this case, we shall just write SA instead of S(A. 

It is the purpose of this (short) section to show that the four conditions given in Theorem 7.3.5 are sufficient to identify a scheme as a semidirect product. 

Let us now see what we can say about characters of semidirect products of S and specific thin schemes. 

We finish this chapter by looking at the case where L is a Coxeter set consisting of two elements and (L) has finite valency and is the kernel of a semidirect product. This is a case which, geometrically, has been investigated by Udo Ott and Stanley Payne. 

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