Semidirect Product Group Research Articles

Black-Scholes Theory and Diffusion Processes on the Cotangent Bundle of the Affine Group.

The Black-Scholes partial differential equation (PDE) from mathematical finance has been analysed extensively and it is well known that the equation can be reduced to a heat equation on Euclidean space by a logarithmic transformation of variables. However, an alternative interpretation is proposed in this paper by reframing the PDE as evolving on a Lie group. This equation can be transformed into a diffusion process and solved using mean and covariance propagation techniques developed previously in the context of solving Fokker–Planck equations on Lie groups. An extension of the Black-Scholes theory with coupled asset dynamics produces a diffusion equation on the affine group, which is not a unimodular group. In this paper, we show that the cotangent bundle of a Lie group endowed with a semidirect product group operation, constructed in this paper for the case of groups with trivial centers, is always unimodular and considering PDEs as diffusion processes on the unimodular cotangent bundle group allows a direct application of previously developed mean and covariance propagation techniques, thereby offering an alternative means of solution of the PDEs. Ultimately these results, provided here in the context of PDEs in mathematical finance may be applied to PDEs arising in a variety of different fields and inform new methods of solution.

A TRINARY RELATION ARISING FROM A MATCHED PAIR OF R-DISCRETE GROUPOIDS

We introduce a notion of a matched pair of r-discrete groupoids and that of a trinary relation associated with a matched pair. We construct three C∗-algebras from a trinary relation and study properties of these algebras. The above results are applied to an action of a countable discrete semidirect product group on a topological space with an invariant measure.

Riesz bases associated with regular representations of semidirect product groups

This work is devoted to the study of Bessel and Riesz systems of the type $$\big \{L_{\gamma }{\mathsf {f}}\big \}_{\gamma \in \Gamma }$$ obtained from the action of the left regular representation $$L_{\gamma }$$ of a discrete non abelian group $$\Gamma $$ which is a semidirect product, on a function $${\mathsf {f}}\in \ell ^2(\Gamma )$$. The main features about these systems can be conveniently studied by means of a simple matrix-valued function $${\mathbf {F}}(\xi )$$. These systems allow to derive sampling results in principal $$\Gamma $$-invariant spaces, i.e., spaces obtained from the action of the group $$\Gamma $$ on a element of a Hilbert space. Since the systems $$\big \{L_{\gamma }{\mathsf {f}}\big \}_{\gamma \in \Gamma }$$ are closely related to convolution operators, a connection with $$C^*$$-algebras is also established.

The inverse monoid associated to a group and the semidirect product of groups

In this paper, we construct an inverse monoid $Mleft( Gright) $ associatedto a given group $G$ by using the notion of the join of subgroups and then,by applying the left action of monoid $M$ on a semigroup $S$, we form asemigroup $Somega M$ on the set $Stimes M$. The finally result is to buildthe semi direct product of groups associated to the group action on ananother group.

Geometry of the Madelung Transform

The Madelung transform is known to relate Schr\"odinger-type equations in quantum mechanics and the Euler equations for barotropic-type fluids. We prove that, more generally, the Madelung transform is a K\"ahler map (i.e. a symplectomorphism and an isometry) between the space of wave functions and the cotangent bundle to the density space equipped with the Fubini-Study metric and the Fisher-Rao information metric, respectively. We also show that Fusca's momentum map property of the Madelung transform is a manifestation of the general approach via reduction for semi-direct product groups. Furthermore, the Hasimoto transform for the binormal equation turns out to be the 1D case of the Madelung transform, while its higher-dimensional version is related to the problem of conservation of the Willmore energy in binormal flows.

Sampling Associated with a Unitary Representation of a Semi-Direct Product of Groups: A Filter Bank Approach

An abstract sampling theory associated with a unitary representation of a countable discrete non abelian group G, which is a semi-direct product of groups, on a separable Hilbert space is studied. A suitable expression of the data samples, the use of a filter bank formalism and the corresponding frame analysis allow for fixing the mathematical problem to be solved: the search of appropriate dual frames for ℓ 2 ( G ) . An example involving crystallographic groups illustrates the obtained results by using either average or pointwise samples.

Quantization of Hamiltonian loop group spaces

We prove a Fredholm property for spin-c Dirac operators $$\mathsf {D}$$ on non-compact manifolds satisfying a certain condition with respect to the action of a semi-direct product group $$K\ltimes \Gamma $$ , with K compact and $$\Gamma $$ discrete. We apply this result to an example coming from the theory of Hamiltonian loop group spaces. In this context we prove that a certain index pairing $$[{\mathcal {X}}] \cap [\mathsf {D}]$$ yields an element of the formal completion $$R^{-\infty }(T)$$ of the representation ring of a maximal torus $$T \subset H$$ ; the resulting element has an additional antisymmetry property under the action of the affine Weyl group, indicating $$[{\mathcal {X}}] \cap [\mathsf {D}]$$ corresponds to an element of the ring of projective positive energy representations of the loop group.

On isometries of symmetric products of metric spaces

By Fn(X), n≥1, we denote the n-th symmetric product of a metric space (X,d) as the space of the nonempty finite subsets of X with at most n elements endowed with the Hausdorff metric dH. By Iso(X) we denote the group of all isometries from X onto itself with the topology of pointwise convergence. In this paper, we show that, under the certain hypothesis, Iso(Fn(X)) is topologically isomorphic to the semidirect product group Iso(Fn(X),F1(X))⋊Iso(X). We apply those results to ℓpq, (p,q)∈[1,∞]×N≥2⁎, as particular spaces and prove the following statements:(1)If p∈{1,∞}, then Iso(F2(ℓp2)) is topologically isomorphic to Z2×Iso(ℓp2).(2)If 3≤q<∞, then Iso(F2(ℓ∞q)) is topologically isomorphic to ∏i=1q−1(Z2)i⋊Iso(ℓ∞q).(3)In other cases except (n,p,q)∈N≥2×{1,∞}×{∞}, the canonical homomorphism χn:Iso(ℓpq)→Iso(Fn(ℓpq)) is a topological isomorphism.

On Generalized Heisenberg Groups: The Symmetric Case

In the literature, the famous Heisenberg group is the group of matrices of the form $$\begin{aligned} \begin{pmatrix} 1 &{}\quad x &{}\quad z\\ 0 &{}\quad 1 &{}\quad y\\ 0 &{}\quad 0 &{}\quad 1 \end{pmatrix}, \end{aligned}$$ where x, y, and z are real numbers. In the present article, we examine a generalized Heisenberg group, obtained from an R-module M endowed with an R-bilinear form $$\beta $$ , where R is a ring with identity. We show that the structure of the generalized Heisenberg group and its generating space are intertwined. In particular, we prove that if $$\beta $$ is symmetric, then the corresponding Heisenberg group possesses an involutive decomposition into subgroups, which eventually becomes the semidirect product of groups. This leads to a better understanding of the algebraic structure of the generalized Heisenberg group as well as its extensions by subgroups.

Quantum localisation on the circle

Covariant integral quantisation using coherent states for semi-direct product groups is implemented for the motion of a particle on the circle. In this case, the phase space is the cylinder, which is viewed as a left coset of the Euclidean group E(2). Coherent states issued from fiducial vectors are labeled by points in the cylinder and depend also on extra parameters. We carry out the corresponding quantisations of the basic classical observables, particularly the angular momentum and the 2π-periodic discontinuous angle function. We compute their corresponding lower symbols. The quantum localisation on the circle is examined through the properties of the angle operator yielded by our procedure, its spectrum and lower symbol, its commutator with the quantum angular momentum, and the resulting Heisenberg inequality. Comparison with other approaches to the long-standing question of the quantum angle is discussed.

Noether's problem on semidirect product groups

Let K be a field, G a finite group. Let G act on the function field L=K(xσ:σ∈G) by τ⋅xσ=xτσ for any σ,τ∈G. Denote the fixed field of the action by K(G)=LG={fg∈L:σ(fg)=fg,∀σ∈G}. Noether's problem asks whether K(G) is rational (purely transcendental) over K.It is known that if G=Cm⋊Cn is a semidirect product of cyclic groups Cm and Cn with Z[ζn] a unique factorization domain, and K contains an eth primitive root of unity, where e is the exponent of G, then K(G) is rational over K.In this paper, we give another criteria to determine whether K(Cm⋊Cn) is rational over K. In particular, if p,q are prime numbers and there exists x∈Z[ζq] such that the norm NQ(ζq)/Q(x)=p, then C(Cp⋊Cq) is rational over C.

Abstract relative Gabor transforms over canonical homogeneous spaces of semidirect product groups with Abelian normal factor

This paper introduces a unified approach to the abstract notion of relative Gabor transforms over canonical homogeneous spaces of semi-direct product groups with Abelian normal factor. Let [Formula: see text] be a locally compact group, [Formula: see text] be a locally compact Abelian (LCA) group, and [Formula: see text] be a continuous homomorphism. Let [Formula: see text] be the semi-direct product of [Formula: see text] and [Formula: see text] with respect to [Formula: see text], [Formula: see text] be the canonical homogeneous space of [Formula: see text], and [Formula: see text] be the canonical relatively invariant measure on [Formula: see text]. Then we present a unified harmonic analysis approach to the theoretical aspects of the notion of relative Gabor transform over the Hilbert function space [Formula: see text].

Ergodic invariant states and irreducible representations of crossed product $C^∗$-algebras

Given a C∗-dynamical system consisting of a unital C∗-algebra A and a discrete group Γ acting on A as automorphisms, we show that the space of (ergodic) Γ-invariant states on A is homeomorphic to a subspace of (pure) state space of A o Γ. It follows that classification of ergodic Γ-invariant regular Borel probability measures on a compact Hausdorff space X is equivalent to classification of irreducible representations of C(X) o Γ whose restriction to Γ contains the trivial representation of Γ. As an application, a reformulation of Furstenberg’s conjecture via representation theory of the semidirect product group Z[ 1 pq ] o Z 2 is obtained.

Study of shearlet transform using block matrix dilation

In this paper, the shearlet theory is extended from ordinary matrices to block matrices and its properties are investigated. More precisely, the block shearlet group is defined as a semidirect product group and the block shearlet transform is constructed by means of a quasiregular representation of this semidirect product group. In addition, we show that the block shearlet transform can overcome two major issues of the classical shearlet transform: time complexity and redundancy. We also report the result of careful experiments on video denoising which compares our proposed method with state of the art multiscale techniques, including curvelets and 3D shearlet transform.

Quaternionic continuous wavelet transform on a quaternionic Hilbert space

In this paper we consider the new concept of the quaternionic unitary representation of a locally compact group to the unitary group of a quaternionic Hilbert space, and study its properties. Also, we establish a continuous wavelet transform by means of a special case of such representations to extend the continuous wavelet transform related to semidirect product groups. The results may be useful in development of wavelet theory.

The Madelung transform as a momentum map

The Madelung transform relates the non-linear Schrödinger equation and a compressible Euler equation known as the quantum hydrodynamical system. We prove that the Madelung transform is a momentum map associated with an action of the semidirect product group $\mathrm{Diff}(\mathbb{R}^{n}) \ltimes H^∞(\mathbb{R}^n; \mathbb{R})$, which is the configuration space of compressible fluids, on the space $Ψ = H^∞(\mathbb{R}^{n}; \mathbb{C})$ of wave functions. In particular, this implies that the Madelung transform is a Poisson map taking the natural Poisson bracket on $Ψ$ to the compressible fluid Poisson bracket. Moreover, the Madelung transform provides an example of 'Clebsch variables' for the hydrodynamical system.

Hamiltonian approach to Ehrenfest expectation values and Gaussian quantum states.

The dynamics of quantum expectation values is considered in a geometric setting. First, expectation values of the canonical observables are shown to be equivariant momentum maps for the action of the Heisenberg group on quantum states. Then, the Hamiltonian structure of Ehrenfest's theorem is shown to be Lie-Poisson for a semidirect-product Lie group, named the Ehrenfest group. The underlying Poisson structure produces classical and quantum mechanics as special limit cases. In addition, quantum dynamics is expressed in the frame of the expectation values, in which the latter undergo canonical Hamiltonian motion. In the case of Gaussian states, expectation values dynamics couples to second-order moments, which also enjoy a momentum map structure. Eventually, Gaussian states are shown to possess a Lie-Poisson structure associated with another semidirect-product group, which is called the Jacobi group. This structure produces the energy-conserving variant of a class of Gaussian moment models that have previously appeared in the chemical physics literature.

Simplified vanishing moment criteria for wavelets over general dilation groups, with applications to abelian and shearlet dilation groups

We consider the coorbit theory associated to a square-integrable, irreducible quasi-regular representation of a semidirect product group G = R d ⋊ H . The existence of coorbit spaces for this very general setting has been recently established, together with concrete vanishing moment criteria for analyzing vectors and atoms that can be used in the coorbit scheme. These criteria depend on fairly technical assumptions on the dual action of the dilation group, and it is one of the chief purposes of this paper to considerably simplify these assumptions. We then proceed to verify the assumptions for large classes of dilation groups, in particular for all abelian dilation groups in arbitrary dimensions, as well as a class called generalized shearlet dilation groups , containing and extending all known examples of shearlet dilation groups employed in dimensions two and higher. We explain how these groups can be systematically constructed from certain commutative associative algebras of the same dimension, and give a full list, up to conjugacy, of shearing groups in dimensions three and four. In the latter case, three previously unknown groups are found. As a result, the existence of Banach frames consisting of compactly supported wavelets, with simultaneous convergence in a whole range of coorbit spaces, is established for all groups involved.

Supramenable groups and partial actions

We characterize supramenable groups in terms of the existence of invariant probability measures for partial actions on compact Hausdorff spaces and the existence of tracial states on partial crossed products. These characterizations show that, in general, one cannot decompose a partial crossed product of a $\text{C}^{\ast }$-algebra by a semidirect product of groups into two iterated partial crossed products. However, we give conditions which ensure that such decomposition is possible.

A new graph over semi-direct products of groups

In this paper, by establishing a new graph ?(G) over the semi-direct product of groups, we will first state and prove some graph-theoretical properties, namely, diameter, maximum and minimum degrees, girth, degree sequence, domination number, chromatic number, clique number of ?(G). In the final section we will show that ?(G) is actually a perfect graph.