Imprimitivity Theorem Research Articles

InducedC *-algebras and a symmetric imprimitivity theorem

InducedC *-algebras and a symmetric imprimitivity theorem

Quantum mechanics of the quaternionic Hilbert spaces based upon the imprimitivity theorem

We survey the realization of quantum mechanics in quaternionic Hilbert spaces following the methods of Mackey, who examined the complex and real cases exploiting the imprimitivity theorem. We show that there exists a unique unitary skew-adjoint operator which commutes with all the observables. This operator not only plays the role of the imaginary unit in the complex case, but allows a complexification of the Hilbert space by the choice of any quaternionic imaginary unit. Difficulties in the definition of time reversal, however, arise because of the properties of the quaternionic field. The introduction of an extra imaginary unit, commuting with the others, is suggested in order to implement time reversal properly. In the Appendix we give the proof of the imprimitivity theorem, in the quaternionic case, that we use in the paper.

Commutative Systems of Covariance and a Generalization of Mackey's Imprimitivity Theorem

AbstractTwo results are obtained in this paper. The first is a generalization of the imprimitivity theorem of Mackey, when the associated projection-valued measure is replaced by a commutative positive operator valued measure. The second is a necessary and sufficient condition for such a system of covariance to possess an overcomplete, covariant family of coherent states.

A mackey imprimitivity theory for algebraic groups

Let G be an affine algebraic group over an algebraically closed field k and let H be a closed subgroup of G. If V is a rational H-module (a comodule for the coordinate ring of H) there is a now well-known notion of an induced module VI G for G, defined as the space Morph/~(G, V) of all H-equivariant morphisms from G to a finite dimensional subspace of V, with obvious G-action. The question arises, given a rational G-module M, how can one recognize M as an induced module VIG? For a finite group G the answer is the Mackey imprimitivity theorem: the module M is induced if and only if it is a direct sum of subspaces permuted transitively by G (with H the stabilizer of one of these subspaces, called V). One uses this result, for example, in proving the famous Mackey decomposition theorem which describes the restriction of any induced module to a second subgroup L as a direct sum of suitable induced modules; given the imprimitivity theorem, the proof is just a matter of grouping the summands permuted by G into their orbits under L. In the case of algebraic groups the situation is quite different. For some subgroups H, all G-modules are induced. This occurs, for example, if G is connected and k [G/H] ,= k, e.g., if H is parabolic. Also, if G is a connected unipotent group, then a rational G-module M is induced from some proper subgroup if and only if its endomorphism ring contains a two dimensional submodule E, for the conjugation action of G, with E___ k. 1 and such that k. l is precisely the subspace annihilated by the action of the Lie algebra of G on E [27] (cf. also (5.5) below). The latter is an application of a general criterion in case G/H is affine: a rational G-module M is induced if and only if there is an action of the coordinate ring A of G/H compatible with the action of G on both A and M. Note that this generalizes the imprimitivity theorem in the case of finite groups, since then A has a k-basis of [G:H] orthogonal idempotents permuted transitively by G.

SHAPE AND INDUCED REPRESENTATIONS

Abstract Let K: P → T be a fixed functor. A criterion is given for a functor M': T → V to be a (right) Kan extension along K of some functor M: P → V. The functors M having a given M' as Kan extension are, in general, classified by continuous functors (V P)o → V. We introduce a notion of system of imprimitivity, generalizing that of Mackey; when the shape category of K is codense in the systems of imprimitivity classify the functors H having M' as Kan extension. As a special case one obtains Mackey's Imprimitivity Theorem for finite groups.

Systems of covariance and subrepresentations of induced representations

In relativistic quantum physics, systems of covariance are used to describe the localizability of massless elementary systems. The notion of a system of covariance generalizes that of a system of imprimitivities by replacing the projection valued measure by a positive operator valued measure. Our main result concerns the constructibility of systems of covariance from systems of imprimitivities on different premises. As a corollary we get an analogous theorem to Mackey's Imprimitivity Theorem characterizing subrepresentations of induced representations.

An imprimitivity theorem for algebraic groups

An imprimitivity theorem for algebraic groups

On Mackey's imprimitivity theorem

On Mackey's imprimitivity theorem

Induced representations and a new proof of the imprimitivity theorem

We present a brief and elementary proof of the Imprimitivity Theorem for an arbitrary locally compact group.

An extended Galilean group and its application to time operators

The representation theory of the Galilean group extended by a one-dimensional group of dilations (which leave invariant the free Schrodinger equation) is studied. The existence of position and time operators is investigated from the standpoint of the imprimitivity theorem, for quantum particles whose states carry a projective representation of the extended group. Position and time operators are shown to exist for some two-component direct sums with non-trivial multipliers and for vector representations with zero helicity.

Group Extensions and Cohomology for Locally Compact Groups. III

We shall define and develop the properties of cohomology groups ${H^n}(G,A)$ which can be associated to a pair (G, A) where G is a separable locally compact group operating as a topological transformation group of automorphisms on the polonais abelian group A. This work extends the results in [29] and [30], and these groups are to be viewed as analogues of the Eilenberg-Mac Lane groups for discrete G and A. Our cohomology groups in dimension one are classes of continuous crossed homomorphisms, and in dimension two classify topological group extensions of G by A. We characterize our cohomology groups in all dimensions axiomatically, and show that two different cochain complexes can be used to construct them. We define induced modules and prove a version of Shapiro’s lemma which includes as a special case the Mackey imprimitivity theorem. We show that the abelian groups ${H^n}(G,A)$ are themselves topological groups in a natural way and we investigate this additional structure.

Group extensions and cohomology for locally compact groups. III

We shall define and develop the properties of cohomology groups H n ( G , A ) {H^n}(G,A) which can be associated to a pair (G, A) where G is a separable locally compact group operating as a topological transformation group of automorphisms on the polonais abelian group A. This work extends the results in [29] and [30], and these groups are to be viewed as analogues of the Eilenberg-Mac Lane groups for discrete G and A. Our cohomology groups in dimension one are classes of continuous crossed homomorphisms, and in dimension two classify topological group extensions of G by A. We characterize our cohomology groups in all dimensions axiomatically, and show that two different cochain complexes can be used to construct them. We define induced modules and prove a version of Shapiro’s lemma which includes as a special case the Mackey imprimitivity theorem. We show that the abelian groups H n ( G , A ) {H^n}(G,A) are themselves topological groups in a natural way and we investigate this additional structure.

A note on Mackey's imprimitivity theorem

A note on Mackey's imprimitivity theorem

Projective unitary antiunitary representations of locally compact groups

We give here a systematic presentation of the theory of projective representations when antiunitary operators are present. In particular the imprimitivity theorem of Mackey is proved in this situation and all the unitary antiunitary representations of the extended Poincare group are derived.

Positive definite measures

In this paper we prove two theorems relating positive definite measures to induced representations. The first shows how the injection of a positive definite measure on a topological group H into a containing locally compact group G in which H is closed gives rise to induced representations. The second is another version of Mackey's imprimitivity theorem, along the lines of Loomis' proof [5]. We feel this is justified on several grounds. Firstly, our proof is simpler than Loomis'. We make no use of the Radon-Nikodym theorem nor of quasi-invariant measures. Secondly, we do not assume in advance that our system of imprimitivity is based on the reduced algebra of Borel sets in G/H. Instead, this fact is seen as a consequence of the theorem. Finally, the statement and proof of Theorem 2 in [5] are in need of minor repairs. Using Loomis' notation, the induced representation space of Vis spanned, not by the set of functions {fu: uCH}, but rather by the set { [E]fu: uCH, E a Borel subset of G/K}. Formula (8) must then be replaced by formula (11) in the statement of the theorem. The algebra Co(SXG) used in the present paper may be looked upon as a device for accomplishing these changes. All nonobvious definitions, notations, and conventions are those of [1i].