Following a general method proposed earlier, we construct here Wigner functions defined on coadjoint orbits of a class of semidirect product groups. The groups in question are such that their unitary duals consist purely of representations from the discrete series and each unitary irreducible representation is associated with a coadjoint orbit. The set of all coadjoint orbits (hence UIRs) is finite and their union is dense in the dual of the Lie algebra. The simple structure of the groups and the orbits enables us to compute the various quantities appearing in the definition of the Wigner function explicitly. Possible use of the Wigner functions so constructed, in image analysis and quantum optical measurements, is suggested.
The semidirect product of a Lie algebra and a 2-term representation up to homotopy is a Lie 2-algebra. Such Lie 2-algebras include many examples arising from the Courant algebroid appearing in generalized complex geometry. In this paper, we integrate such a Lie 2-algebra to a strict Lie 2-group in the finite dimensional case.
One of pressing problems in mathematical physics is to find a generalized Poincaré symmetry that could be applied to nonflat space-times. As a step in this direction we define the semidirect product of groupoids $Γ_0 \rtimes Γ_1$ and investigate its properties. We also define the crossed product of a bundle of algebras with the groupoid $Γ_1$ and prove that it is isomorphic to the convolutive algebra of the groupoid $Γ_0 \rtimes Γ_1$. We show that families of unitary representations of semidirect product groupoids in a bundle of Hilbert spaces are random operators. An important example is the …
Here we carry out computations that help clarify the Lagrangian and Hamiltonian structure of compressible flow. The intent is to be pedagogical and rigorous, providing concrete examples of the theory outlined in Holm, Marsden, and Ratiu [1998] and Marsden, Ratiu, and Weinstein [1984].
We study integrable systems on double Lie algebras in absence of Ad-invariant bilinear form by passing to the semidirect product with the $τ$-representation. We show that in this stage a natural Ad-invariant bilinear form does exist, allowing for a straightforward application of the AKS theory, and giving rise to Manin triple structure, thus bringing the problem to the realm of Lie bialgebras and Poisson-Lie groups.
The Landau-Lifshitz equation is derived as the reduction of a geodesic flow on the group of maps into the rotation group. Passing the symmetries of spatial isotropy to the reduced space is an example of semidirect product reduction by stages.
We study integrable systems on the semidirect product of a Lie group and its Lie algebra as the representation space of the adjoint action. Regarding the tangent bundle of a Lie group as phase space endowed with this semidirect product Lie group structure, we construct a class of symplectic submanifolds equipped with a Dirac bracket on which integrable systems (in the Adler-Kostant-Symes sense) are naturally built through collective dynamics. In doing so, we address other issues as factorization, Poisson-Lie structures and dressing actions. We show that the procedure becomes recursive for some…
Let $H$ be a locally compact group and $K$ be an LCA group also let $τ:H\to Aut(K)$ be a continuous homomorphism and $G_τ=H\ltimes_τK$ be the semidirect product of $H$ and $K$ with respect to $τ$. In this article we define the Zak transform $\mathcal{Z}_L$ on $L^2(G_τ)$ with respect to a $τ$-invariant uniform lattice $L$ of $K$ and we also show that the Zak transform satisfies the Plancherel formula. As an application we show that how these techniques apply for the semidirect product group $\mathrm{SL}(2,\mathbb{Z})\ltimes_τ\mathbb{R}^2$ and also the Weyl-Heisenberg groups.
Continuous wavelet transforms arising from the quasiregular representation of a semidirect product of a vector group with a matrix group -- the so-called dilation group -- have been studied by various authors. Recently the attention has shifted from the irreducible case to include more general dilation groups, for instance cyclic (more generally: discrete) or one-parameter groups. These groups do not give rise to irreducible square-integrable representations, yet it is possible (and quite simple) to give admissibility conditions for a large class of them. We put these results in a theoretical …
Given a semidirect product $\frak{g}=\frak{s}\uplus\frak{r}$ of semisimple Lie algebras $\frak{s}$ and solvable algebras $\frak{r}$, we construct polynomial operators in the enveloping algebra $\mathcal{U}(\frak{g})$ of $\frak{g}$ that commute with $\frak{r}$ and transform like the generators of $\frak{s}$, up to a functional factor that turns out to be a Casimir operator of $\frak{r}$. Such operators are said to generate a virtual copy of $\frak{s}$ in $\mathcal{U}(\frak{g})$, and allow to compute the Casimir operators of $\frak{g}$ in closed form, using the classical formulae for the invaria…
In this paper, we study two-component versions of the periodic Hunter-Saxton equation and its $μ$-variant. Considering both equations as a geodesic flow on the semidirect product of the circle diffeomorphism group $\Diff(§)$ with a space of scalar functions on $§$ we show that both equations are locally well-posed. The main result of the paper is that the sectional curvature associated with the 2HS is constant and positive and that 2$μ$HS allows for a large subspace of positive sectional curvature. The issues of this paper are related to some of the results for 2CH and 2DP presented in [J. Esc…
Yang-Mills theory with a symmetry algebra that is the semidirect product $\mathfrak{h}\ltimes\mathfrak{h}^*$ defined by the coadjoint action of a Lie algebra $\mathfrak{h}$ on its dual $\mathfrak{h}^*$ is studied. The gauge group is the semidirect product ${\rm G}_{\mathfrak{h}}\ltimes{\mathfrak{h}^*}$, a noncompact group given by the coadjoint action on $\mathfrak{h}^*$ of the Lie group ${\rm G}_{\mathfrak{h}}$ of $\mathfrak{h}$. For $\mathfrak{h}$ simple, a method to construct the self-antiself dual instantons of the theory and their gauge non\-equivalent deformations is presented. Every ${\…
A discrete version of Lagrangian reduction is developed in the context of discrete time Lagrangian systems on $G\times G$, where $G$ is a Lie group. We consider the case when the Lagrange function is invariant with respect to the action of an isotropy subgroup of a fixed element in the representation space of $G$. In this context the reduction of the discrete Euler-Lagrange equations is shown to lead to the so called discrete Euler-Poincaré equations. A constrained variational principle is derived. The Legendre transformation of the discrete Euler-Poincaré equations leads to discrete Hamiltoni…
This paper develops the theory of Dirac reduction by symmetry for nonholonomic systems on Lie groups with broken symmetry. The reduction is carried out for the Dirac structures, as well as for the associated Lagrange-Dirac and Hamilton-Dirac dynamical systems. This reduction procedure is accompanied by reduction of the associated variational structures on both Lagrangian and Hamiltonian sides. The reduced dynamical systems obtained are called the implicit Euler-Poincaré-Suslov equations with advected parameters and the implicit Lie-Poisson-Suslov equations with advected parameters. The theory …
In the usual formulation of quantum mechanics, groups of automorphisms of quantum states have ray representations by unitary and antiunitary operators on complex Hilbert space, in accordance with Wigner's Theorem. In the phase-space formulation, they have real, true unitary representations in the space of square-integrable functions on phase-space. Each such phase-space representation is a Weyl-Wigner product of the corresponding Hilbert space representation with its contragredient, and these can be recovered by `factorising' the Weyl-Wigner product. However, not every real, unitary representa…
We develop a pseudo-differential Weyl calculus on nilpotent Lie groups which allows one to deal with magnetic perturbations of right invariant vector fields. For this purpose we investigate an infinite-dimensional Lie group constructed as the semidirect product of a nilpotent Lie grup and an appropriate function space thereon. We single out an appropriate coadjoint orbit in the semidirect product and construct our pseudo-differential calculus as a Weyl quantization of that orbit.
Born proposed a unification of special relativity and quantum mechanics that placed position, time, energy and momentum on equal footing through a reciprocity principle and extended the usual position-time and energy-momentum line elements to this space by combining them through a new fundamental constant. Requiring also invariance of the symplectic metric yields U(1,3) as the invariance group, the inhomogeneous counterpart of which is the canonically relativistic group CR(1,3) = U(1,3) *s H(1,3) where H(1,3) is the Heisenberg Group in 4 dimensions and "*s" is the semidirect product. This is t…
We present a generalization of Wigner's unitary-antiunitary theorem for pairs of ray transformations. As a particular case, we get a new Wigner-type theorem for non-Hermitian indefinite inner product spaces.
In this paper, we introduce the notion of generalized representation of a $3$-Lie algebra, by which we obtain a generalized semidirect product $3$-Lie algebra. Moreover, we develop the corresponding cohomology theory. Various examples of generalized representations of 3-Lie algebras and computation of 2-cocycles of the new cohomology are provided. Also, we show that a split abelian extension of a 3-Lie algebra is isomorphic to a generalized semidirect product $3$-Lie algebra. Furthermore, we describe general abelian extensions of 3-Lie algebras using Maurer-Cartan elements.
A symmetry in quantum mechanics is described by the projective representations of a Lie symmetry group that transforms between physical quantum states such that the square of the modulus of the states is invariant. The Heisenberg commutation relations, that are fundamental to quantum mechanics, must be valid in all of these physical states. This paper shows that the maximal quantum symmetry group, whose projective representations preserve the Heisenberg commutation relations in this manner, is the inhomogeneous symplectic group. The projective representations are equivalent to the unitary repr…
