Pose Changes From a Different Point of View

For more than a century, rigid-body displacements have been viewed as affine transformations described as homogeneous transformation matrices wherein the linear part is a rotation matrix. In group-theoretic terms, this classical description makes rigid-body motions a semidirect product. The distinction between a rigid-body displacement of Euclidean space and a change in pose from one reference frame to another is usually not articulated well in the literature. Here, we show that, remarkably, when changes in pose are viewed from a space-fixed reference frame, the space of pose changes can be endowed with a direct product group structure, which is different from the semidirect product structure of the space of motions. We then show how this new perspective can be applied more naturally to problems such as monitoring the state of aerial vehicles from the ground, or the cameras in a humanoid robot observing pose changes of its hands.

Pose Changes From a Different Point of View

For more than a century, rigid-body displacements have been viewed as affine transformations described as homogeneous transformation matrices wherein the linear part is a rotation matrix. In group-theoretic terms, this classical description makes rigid-body motions a semi-direct product. The distinction between a rigid-body displacement of Euclidean space and a change in pose from one reference frame to another is usually not articulated well in the literature. Here we show that, remarkably, when changes in pose are viewed from a space-fixed reference frame, the space of pose changes can be endowed with a direct product group structure, which is different from the semi-direct product structure of the space of motions. We then show how this new perspective can be applied more naturally to problems such as monitoring the state of aerial vehicles from the ground, or the cameras in a humanoid robot observing motions of its hands.

Definition and Properties of Direct Sum Decomposition of Groups

Summary In this article, direct sum decomposition of group is mainly discussed. In the second section, support of element of direct product group is defined and its properties are formalized. It is formalized here that an element of direct product group belongs to its direct sum if and only if support of the element is finite. In the third section, product map and sum map are prepared. In the fourth section, internal and external direct sum are defined. In the last section, an equivalent form of internal direct sum is proved. We referred to [23], [22], [8] and [18] in the formalization.

MULTIPLICITY-FREE DECOMPOSITIONS OF THE MINIMAL REPRESENTATION OF THE INDEFINITE ORTHOGONAL GROUP

Kazhdan, Kostant, Binegar–Zierau and Kobayashi–Ørsted constructed a distinguished infinite-dimensional irreducible unitary representation π of the indefinite orthogonal group G = O(2p, 2q) for p, q ≥ 1 with p + q > 2, which has the smallest Gelfand–Kirillov dimension 2p + 2q - 3 among all infinite-dimensional irreducible unitary representations of G and hence is called the minimal representation. We consider, for which subgroup G′ of G, the restriction π|G′ is multiplicity-free. We prove that the restriction of π to any subgroup containing the direct product group U(p1) × U(p2) × U(q) for p1, p2 ≥ 1 with p1 + p2 = p is multiplicity-free, whereas the restriction to U(p1) × U(p2) × U(q1) × U(q2) for q1, q2 ≥ 1 with q1 + q2 = q has infinite multiplicities.

Cylindrical representations of some infinite dimensional nuclear Lie groups

Let Γ.𝒜 be the semi-direct product group of a nuclear Lie group Γ with the additive group 𝒜 of a real nuclear vector space. We give an explicit description of all the continuous representations of Γ.𝒜 the restriction of which to 𝒜 is a cyclic unitary representation, and a necessary and sufficient condition for the unitarity of such cylindrical representations is stated. This general result is successfully used to obtain irreducible unitary representations of the nuclear Lie groups of Riemannian motions, and, in the setting of the theory of multiplicative distributions initiated by I.M. Gelfand, it is proved that for any connected real finite dimensional Lie groupGand for any strictly positive integerkthere exist non located and non trivially decomposable representations of orderkof the nuclear Lie group(M;G) of all theG-valued test-functions on a given paracompact manifoldM.

A dynamical group for the harmonic oscillator

A dynamical group is constructed for the isotropic three-dimensional harmonic oscillator by forming the semi-direct product group W (3)⊗ Sp (6, R ), where W (3) is the Weyl group and Sp (6, R ) the real symplectic group. A single representation of W (3)⊗ Sp (6, R ) is con­structed, using the usual harmonic oscillator annihilation and creation operators for W (3) and their anticommutators for Sp (6, R ), which can be spanned by the complete set of harmonic oscillator states. The group W (3)⊗ Sp (6, R ) is found to simplify the calculation of matrix elements of operators acting on harmonic oscillator states and to have a rich and useful subgroup structure.