Eldar Straume

Compact connected Lie transformation groups on spheres with low cohomogeneity. II

Organization of orthogonal models and orbit structures Orbit structures for G-spheres of cohomogeneity two The reconstruction problem G-spheres of cohomogeneity two with at most two isolated orbits G-spheres of cohomogeneity two with three isolated orbits Figures References.

Differential Equations - Geometry, Symmetries and Integrability

This article reviews some recent theoretical results about the structure of Darboux integrable differential systems and their relationship with symmetry reduction of exterior differential systems. The symmetry reduction representation of Darboux integrable equations is then used to derive some new and unusual transformations.

On the geometry and behavior of n-body motions

The kinematic separation of size, shape, and orientation of n-body systems is investigated together with specific issues concerning the dynamics of classical n-body motions. A central topic is the asymptotic behavior of general collisions, extending the early work of Siegel, Wintner, and more recently Saari. In particular, asymptotic formulas for the derivatives of any order of the basic kinematic quantities are included. The kine- matic Riemannian metric on the congruence and shape moduli spaces are introduced via O(3)-equivariant geometry. For n = 3, a classical geometrization procedure is explicitly carried out for planary 3-body motions, reducing them to solutions of a rather simple system of geodesic equations in the 3-dimensional congruence space. The paper is largely expository and various known results on classical n-body motions are surveyed in our more geometrical setting.

Cohomogeneity one anti de Sitter space AdSn+1

In this paper we study the anti de Sitter space AdSn+1 under a cohomogeneity one action of a connected closed Lie subgroup G of the isometry group. Among various results, for compact groups we determine the possible acting groups, the orbit space and principal and singular orbits. For noncompact groups it is shown that if there is a principal orbit which is either simply connected or totally umbilic, then there is only one orbit type. Furthermore, in the totally umbilic case, all orbits are congruent to AdSn.

Symbol correspondences for spin systems

Preface.- 1 Introduction.- 2 Preliminaries.- 3 Quantum Spin Systems and Their Operator Algebras.- 4 The Poisson Algebra of the Classical Spin System.- 5 Intermission.- 6 Symbol Correspondences for a Spin-j System.- 7 Multiplications of Symbols on the 2-Sphere.- 8 Beginning Asymptotic Analysis of Twisted Products.- 9 Conclusion.- Appendix.- Bibliography.- Index.

Quantum Spin Systems and Their Operator Algebras

This chapter presents the basic mathematical framework for quantum mechanics of spin systems. Much of the material can be found in texts in representation theory (some found within the list of references at the beginning of Chap. 2) and quantum theory of angular momentum (e.g. [13, 14, 16, 23, 46, 63, 65], some of these being textbooks in quantum mechanics which can also be used by the reader not too familiar with the subject as a whole). Our emphasis here is to provide a self-contained presentation of quantum spin systems where, in particular, the combinatorial role of Clebsch-Gordan coefficients and various kinds of Wigner symbols is elucidated, leading to the SO(3)-invariant decomposition of the operator product which, strangely enough, we have not found explicitly done anywhere.

The Poisson Algebra of the Classical Spin System

This chapter presents the basic mathematical framework for classical mechanics of a spin system. Practically all the material in the introductory section below can be found in basic textbooks on classical mechanics and we refer to some of these, e.g. [1, 5, 34, 37, 48], for the reader not yet too familiar with the subject, or for further details, examples, etc. (Ref. [34] is more familiar to physicists, while the others are more mathematical and closer in style to our brief introduction below). Our emphasis here is to provide a self-contained presentation of the SO(3)-invariant decomposition of the pointwise product and the Poisson bracket of polynomials, which are not easily found elsewhere (specially the latter).

Symbol Correspondences for a Spin-j System

In this chapter we define, classify and study symbol correspondences for a spin-j system, presenting explicit constructions. Our cornerstone is the concept of characteristic numbers of a symbol correspondence, which provides coordinates on the moduli space of spin-j symbol correspondences. As we shall see below, for any j a (quite smaller) subset of characteristic numbers can be distinguished in terms of a stricter requirement for an isometric correspondence. However, a more subtle distinction is obtained in the asymptotic limit n = 2j → ∞, to be explored in Chap. 8.

Appendix: Further Proofs

In this Appendix we gather proofs of some of the propositions and a theorem, which were stated in the main text.

Multiplications of Symbols on the 2-Sphere

Given any symbol correspondence \(W^{j} = W_{\vec{c}}^{j}\), the algebra of operators in \(\mathcal{B}(\mathcal{H}_{j}) \simeq M_{\mathbb{C}}(n + 1)\) can be imported to the space of symbols \(W_{\vec{c}}^{j}(\mathcal{B}(\mathcal{H}_{j})) \simeq \mathit{Poly}_{\mathbb{C}}(S^{2})_{\leq n} \subset C_{\mathbb{C}}^{\infty }(S^{2})\). The 2-sphere, with such an algebra on a subset of its function space, has become known as the “fuzzy sphere” [47]. However, there is no single “fuzzy sphere”, as each symbol correspondence defined by characteristic numbers \(\vec{c} = (c_{1},\ldots,c_{n})\) gives rise to a distinct (although isomorphic) algebra on the space of symbols \(\mathit{Poly}_{\mathbb{C}}(S^{2})_{\leq n}\), as we shall investigate in some detail, in this chapter.