Abstract: This paper reviews the properties and applications of certain n-ary generalizations of Lie algebras in a self-contained and unified way. These generalizations are algebraic structures in which the two-entry Lie bracket has been replaced by a bracket with n entries. Each type of n-ary bracket satisfies a specific characteristic identity which plays the role of the Jacobi identity for Lie algebras. Particular attention will be paid to generalized Lie algebras, which are defined by even multibrackets obtained by antisymmetrizing the associative products of its n components and that satisfy the generalized Jacobi identity, and to Filippov (or n-Lie) algebras, which are defined by fully antisymmetric n-brackets that satisfy the Filippov identity. 3-Lie algebras have surfaced recently in multi-brane theory in the context of the Bagger-Lambert-Gustavsson model. As a result, Filippov algebras will be discussed at length, including the cohomology complexes that govern their central extensions and their deformations ( it turns out that Whitehead's lemma extends to all semisimple n-Lie algebras). When the skewsymmetry of the Lie or n-Lie algebra bracket is relaxed, one is led to a more general type of n-algebras, the n-Leibniz algebras. These will be discussed as well, since they underlie the cohomological properties of n-Lie algebras. The standard Poisson structure may also be extended to the n-ary case. We shall review here the even generalized Poisson structures, whose generalized Jacobi identity reproduces the pattern of the generalized Lie algebras, and the Nambu-Poisson structures, which satisfy the Filippov identity and determine Filippov algebras. Finally, the recent work of Bagger-Lambert and Gustavsson on superconformal Chern-Simons theory will be briefly discussed. Emphasis will be made on the appearance of the 3-Lie algebra structure and on why the A(4) model may be formulated in terms of an ordinary Lie algebra, and on its Nambu bracket generalization.

Abstract: Nuclear ground state properties including mass, charge radii, spins and moments can be determined by applying atomic physics techniques such as Penning-trap based mass spectrometry and laser spectroscopy. The MATS and LaSpec setups at the low-energy beamline at FAIR will allow us to extend the knowledge of these properties further into the region far from stability. The mass and its inherent connection with the nuclear binding energy is a fundamental property of a nuclide, a unique “fingerprint”. Thus, precise mass values are important for a variety of applications, ranging from nuclear-structure studies like the investigation of shell closures and the onset of deformation, tests of nuclear mass models and mass formulas, to tests of the weak interaction and of the Standard Model. The required relative accuracy ranges from 10(-5) to below 10(-8) for radionuclides, which most often have half-lives well below 1 s. Substantial progress in Penning trap mass spectrometry has made this method a prime choice for precision measurements on rare isotopes. The technique has the potential to provide high accuracy and sensitivity even for very short-lived nuclides. Furthermore, ion traps can be used for precision decay studies and offer advantages over existing methods. With MATS (Precision Measurements of very short-lived nuclei using an Advanced Trapping System for highly-charged ions) at FAIR we aim to apply several techniques to very short-lived radionuclides: High-accuracy mass measurements, in-trap conversion electron and alpha spectroscopy, and trap-assisted spectroscopy. The experimental setup of MATS is a unique combination of an electron beam ion trap for charge breeding, ion traps for beam preparation, and a high-precision Penning trap system for mass measurements and decay studies. For the mass measurements, MATS offers both a high accuracy and a high sensitivity. A relative mass uncertainty of 10(-9) can be reached by employing highly-charged ions and a non-destructive Fourier-Transform Ion-Cyclotron-Resonance (FT-ICR) detection technique on single stored ions. This accuracy limit is important for fundamental interaction tests, but also allows for the study of the fine structure of the nuclear mass surface with unprecedented accuracy, whenever required. The use of the FT-ICR technique provides true single ion sensitivity. This is essential to access isotopes that are produced with minimum rates which are very often the most interesting ones. Instead of pushing for highest accuracy, the high charge state of the ions can also be used to reduce the storage time of the ions, hence making measurements on even shorter-lived isotopes possible. Decay studies in ion traps will become possible with MATS. Novel spectroscopic tools for in-trap high-resolution conversion-electron and charged-particle spectroscopy from carrier-free sources will be developed, aiming e. g. at the measurements of quadrupole moments and E0 strengths. With the possibility of both high-accuracy mass measurements of the shortest-lived isotopes and decay studies, the high sensitivity and accuracy potential of MATS is ideally suited for the study of very exotic nuclides that will only be produced at the FAIR facility. Laser spectroscopy of radioactive isotopes and isomers is an efficient and model-independent approach for the determination of nuclear ground and isomeric state properties. Hyperfine structures and isotope shifts in electronic transitions exhibit readily accessible information on the nuclear spin, magnetic dipole and electric quadrupole moments as well as root-mean-square charge radii. The dependencies of the hyperfine splitting and isotope shift on the nuclear moments and mean square nuclear charge radii are well known and the theoretical framework for the extraction of nuclear parameters is well established. These extracted parameters provide fundamental information on the structure of nuclei at the limits of stability. Vital information on both bulk and valence nuclear properties are derived and an exceptional sensitivity to changes in nuclear deformation is achieved. Laser spectroscopy provides the only mechanism for such studies in exotic systems and uniquely facilitates these studies in a model-independent manner. The accuracy of laser-spectroscopic-determined nuclear properties is very high. Requirements concerning production rates are moderate; collinear spectroscopy has been performed with production rates as few as 100 ions per second and laser-desorption resonance ionization mass spectroscopy (combined with beta-delayed neutron detection) has been achieved with rates of only a few atoms per second. This Technical Design Report describes a new Penning trap mass spectrometry setup as well as a number of complementary experimental devices for laser spectroscopy, which will provide a complete system with respect to the physics and isotopes that can be studied. Since MATS and LaSpec require high-quality low-energy beams, the two collaborations have a common beamline to stop the radioactive beam of in-flight produced isotopes and prepare them in a suitable way for transfer to the MATS and LaSpec setups, respectively.