Motions in Taub-NUT–de Sitter spinning spacetime

We investigate the geodesic motion of pseudo-classical spinning particles in the Taub-NUT–de Sitter spacetime. We obtain the conserved quantities from the solutions of the generalized Killing equations for spinning spaces. Applying the formalism the motion of a pseudo-classical Dirac fermion is analyzed on a cone and plane.     

Symmetries and Motions in NUT-Taub Spinning Space

We review the geodesic motion of pseudo-classical spinning particles in curved space. We describe the generalized Killing equations for spinning spaces and express the constants of motion. We apply the formalism to solve for the motion of a pseudo-classical Dirac fermion in NUT-Taub spinning space and analyze the motion on a cone and on a plane.     

Supersymmetries and Constants of Motion in Taub-NUT Spinning Space

We review the geodesic motion of pseudo-classical spinning particles in curved spaces. Investigating the generalized Killing equations for spinning spaces, we express the constants of motion in terms of Killing-Yano tensors. The general results are applied to the case of the four-dimensional Euclidean Taub-NUT spinning space. A simple exact solution, corresponding to trajectories lying on a cone, is given.     

Geodesic Motion on Extended Taub-NUT Spinning Space

In this paper we investigate the geodesic motion of the pseudo-classical spinning particle for the extended Taub-NUT metric. The generalized equations for spinning space are investigated and the constants of motion are derived in terms of the solutions of these equations. We find only two types of extended Taub-NUT metrics with Kepler type symmetry admitting Killing-Yano tensors. The solutions for the lowest components of generalized Killing equations are presented for a particular form of extended Taub-NUT metric.     

Separability structures and Killing-Yano tensors in vacuum type-D space-times without acceleration

Relationships among the existence of Killing tensors, Killing-Yano tensors, and separability structures with two Killing vectors in vacuum type-D space times are investigated. It is proved that the existence of those objects is equivalent with the assumption that space-time is without acceleration.Work partially supported by GNFM of Italian National Research Council and a Polish Interdisciplinary Research Project MR-I-7.     

Hidden symmetries and killing tensors on curved spaces

Higher-order symmetries corresponding to Killing tensors are investigated. The intimate relation between Killing-Yano tensors and nonstandard supersymmetries is pointed out. In the Dirac theory on curved spaces, Killing-Yano tensors generate Dirac-type operators involved in interesting algebraic structures as dynamical algebras or even infinite dimensional algebras or superalgebras. The general results are applied to space-times which appear in modern studies. One presents the infinite dimensional superalgebra of Dirac type operators on the 4-dimensional Euclidean Taub-NUT space that can be seen as a twisted loop algebra. The existence of the conformal Killing-Yano tensors is investigated for some spaces with mixed 3-Sasakian structures.     

Quasi-Hamiltonian mechanics associated to representations of Lie algebras

For relative classical mechanics we construct a quasi-hamiltonian formalism associated to special representations of Lie algebras. This formalism is natural. For the case of the adjoint representation, this construction reduces to the usual absolute hamiltonian formalism on the dual spaces to Lie algebras.     

Generalised bosonic construction of Virasoro algebras

Energy-momentum tensors of conformal field theories and some of their primary fields, including those of parafermionic theories based on simply-laced Lie algebras, are constructed from free bosons. The classification of such theories requires a generalisation of the root systems of Lie algebras. The complete list of such energy-momentum tensors, that can be constructed from two free bosons, includes those of the first four c<1 theories.     

Projective differential geometry and geodesic conservation laws in general relativity,II: Conservation laws

We examine the geodesic conservation laws associated with the projective actions discussed in our earlier paper with the same overall title. Using the Cartan formalism, a one-to-one correspondence between a class of these actions and all geodesic conservation laws is possible. In particular there is a natural geometric interpretation of Killing tensors. Homothetic motions are shown to correspond to conserved quantities on all geodesies (not just null ones). The same approach identifies homothetic Killing tensors and a universal quadratic first integral which reduces to the conformai Killing tensor case on null geodesics.     

Third-order Killing tensors in the Schwarzschild space-time

The general solution for the third-order Killing tensor equation in the Schwarzschild space-time is written down. It follows that the Schwarzschild metric admits only redundant Killing tensors of order 3.This work was carried out under the auspices of the National Group for Mathematical Physics of C. N. R.     

Structural parameters of semi-simple lie algebras

Based on the exploitations of properties of the Killing forms of semi-simple Lie algebras, we set out in a readily programmable form, the structural analysis and the Iwasawa-type decompositions of semi-simple Lie algebras. As an example, the case ofSO(3,1) and its covering groupSL(2,C) is worked out in some detail.     

(Conformal) Killing Vectors in the Newman-Penrose Formalism

Finding (conformal) Killing vectors of a given metric can be a difficult task. This paper presents an efficient technique for finding Killing, homothetic, or even proper conformal Killing vectors in the Newman-Penrose (NP) formalism. Leaning on, and extending, results previously derived in the GHP formalism we show that the (conformal) Killing equations can be replaced by a set of equations involving the commutators of the Lie derivative with the four NP differential operators, applied to the four coordinates.It is crucial that these operators refer to a preferred tetrad relative to the (conformal) Killing vectors, a notion to be defined. The equations can then be readily solved for the Lie derivative of the coordinates, i.e. for the components of the (conformal) Killing vectors. Some of these equations become trivial if some coordinates are chosen intrinsically (where possible), i.e. if they are somehow tied to the Riemann tensor and its covariant derivatives.If part of the tetrad, i.e. part of null directions and gauge, can be defined intrinsically then that part is generally preferred relative to any Killing vector. This is also true relative to a homothetic vector or a proper conformal Killing vector provided we make a further restriction on that intrinsic part of the tetrad. If because of null isotropy or gauge isotropy, where part of the tetrad cannot even in principle be defined intrinsically, the tetrad is defined only up to (usually) one null rotation parameter and/or a gauge factor, then the NP-Lie equations become slightly more involved and must be solved for the Lie derivative of the null rotation parameter and/or of the gauge factor as well. However, the general method remains the same and is still much more efficient than conventional methods.Several explicit examples are given to illustrate the method.     

Higher-Order Simple Lie Algebras

It is shown that the non-trivial cocycles on simple Lie algebras may be used to introduce antisymmetric multibrackets which lead to higher-order Lie algebras, the definition of which is given. Their generalised Jacobi identities turn out to be satisfied by the antisymmetric tensors (or higher-order “structure constants”) which characterise the Lie algebra cocycles. This analysis allows us to present a classification of the higher-order simple Lie algebras as well as a constructive procedure for them. Our results are synthesised by the introduction of a single, complete BRST operator associated with each simple algebra. Received: 3 June 1996 / Accepted: 8 November 1996     

LETTER: Note on Killing Yano Tensors Admitted by Spherically Symmetric Static Space-Times

The Killing Yano tensors of order two admitted by a general class of spherically symmetric static space-times are found. All such space-times admit at least one Killing Yano tensor and four special cases exist, one admitting four Killing Yano tensors the others admitting ten Killing Yano tensors. The Killing Yano tensors are used to construct second order non-stationary Killing tensors and the nature of the redundancy of these Killing tensors is discussed with reference to the time dependence of the constituent tensors/vectors.     

Invariant differential operators for non-compact Lie groups: The main su(n, n) cases

In the present paper we continue the project of systematic construction of invariant differential operators on the example of the non-compact algebras su(n, n). Our choice of these algebras is motivated by the fact that for n = 2 this is the conformal algebra of 4-dimensional Minkowski space-time. Furthermore for general n these algebras belong to a narrow class of algebras, which we call “conformal Lie algebras”, which have very similar properties to the conformal algebras of n ²-dimensional Minkowski space-time. We give the main multiplets of indecomposable elementary representations for n = 2, 3, 4, including the necessary data for all relevant invariant differential operators.     

Free differential algebras: Their use in field theory and dual formulation

The gauging of free differential algebras (FDA's) produces gauge field theories containing antisymmetric tensors. The FDA's extend the Cartan-Maurer equations of ordinary Lie algebras by incorporating p-form potentials (p>1). We study here the algebra of FDA transformations. To every p-form in the FDA, we associate an extended Lie derivative l generating a corresponding gauge transformation. The field theory based on the FDA is invariant under these new transformations. This gives geometrical meaning to the antisymmetric tensors. The algebra of Lie derivatives is shown to close and provides the dual formulation of FDA's.     

Special tensors in the deformation theory of quadratic algebras for the classical Lie algebras

Using deformation theory, Braverman and Joseph constructed certain primitive ideals in the enveloping algebras of the simple Lie algebras. Except in the case sl(2,C)$\mathfrak{s}\mathfrak{l}(2,\mathbb{C})$, there is a special value of the deformation parameter giving an ideal of infinite codimension. For the classical Lie algebras, the uniqueness of the special value is equivalent to the existence of tensors with very particular properties. The existence of these tensors was concluded abstractly by Braverman and Joseph but here we present explicit formulae. This allows a rather direct computation of the special value of the deformation parameter.     

Spinning Particles in Reissner-Nordström-de Sitter Spacetime

We study the geodesic motion of pseudo-classical spinning particles in the Reissner-Nordström-de Sitter spacetime. We investigate the generalized Killing equations for spinning space and derive the constants of motion in terms of the solutions of these equations. We discuss bound state orbits in a plane.     

Classification of Cylindrically Symmetric Static Spacetimes According to Their Ricci Collineations

A complete classification of cylindrically symmetric static Lorentzian manifolds according to their Ricci collineations (RCs) is provided. The Lie algebras of RCs for the non-degenerate Ricci tensor have dimensions 3 to 10, excluding 8 and 9. For the degenerate tensor the algebra is mostly but not always infinite dimensional; there are cases of 10-, 5-, 4- and 3-dimensional algebras. The RCs are compared with the Killing vectors (KVs) and homothetic motions (HMs). The (non-linear) constraints corresponding to the Lie algebras are solved to construct examples which include some exact solutions admitting proper RCs. Their physical interpretation is given. The classification of plane symmetric static spacetimes emerges as a special case of this classification when the cylinder is unfolded.     

Invariant tensors for simple groups

The forms of the invariant primitive tensors for the simple Lie algebras A_l, B_l, C_l, and D_l are investigated. A new family of symmetric invariant tensors is introduced using the non-trivial cocycles for the Lie algebra cohomology. For the A_l algebra it is explicitly shown that the generic forms of these tensors become zero except for the l primitive ones and that they give rise to the l primitive Casimir operators. Some recurrence and duality relations are given for the Lie algebra cocycles. Tables for the 3- and 5-cocycles for su(3) are su(4) are also provided. Finally, new relations involving the d and f su(n) tensors are given.