Underlying Lie Group Research Articles

D-branes in the diagonal SU(2) coset

The symmetry preserving D-branes in coset theories have previously been described as being centered around projections of products of conjugacy classes in the underlying Lie groups. Here, we investigate the coset where a diagonal action of SU(2) is divided out from SU(2) × SU(2). The corresponding target space is described as a (3-dimensional) pillow with four distinguished corners. It is shown that the (fractional) brane which corresponds to the fixed point that arises in the CFT description, is spacefilling. Moreover, the spacefilling brane is the only one that reaches all of the corners. The other branes are 3, 1 and 0 - dimensional.

Lie supergroups supported over [formula omitted] and [formula omitted] associated to the adjoint representation

The algebraic classification of Lie superalgebras based on g l 2 whose odd module is g l 2 itself under the adjoint representation, together with the existence and uniqueness theorem for ODE’s in supermanifolds, are hereby used as in Lie’s theory to produce nonisomorphic Lie supergroups, all of which are supported over the same underlying Lie group GL 2 , and whose Lie superalgebras of left-invariant supervector fields are isomorphic to those given by the algebraic classification. Their compact real forms are also studied so as to produce nonisomorphic Lie supergroups supported over the unitary group U 2 , and their corresponding maximal tori are described.

Small time controllability of systems on compact Lie groups and spin angular momentum

In this article, we develop some general results on the properties of the reachable sets for right invariant bilinear systems with state varying on compact Lie groups. The main results consist of a characterization of the set of states reachable in arbitrary time from the identity of the group. This, under suitable assumptions, is proved to be a Lie subgroup of the underlying Lie group. We apply these results to the analysis of the controllability of particles with spin. The results are motivated by and generalize the results in another work [D. D’Alessandro, Sys. Control Lett. 41, 213–221 (2000)], where the specific model of a spin 12 particle system in an electro-magnetic field was considered.

Minimal Representations, Spherical Vectors¶and Exceptional Theta Series

Theta series for exceptional groups have been suggested as a possible description of the eleven-dimensional quantum supermembrane. We present explicit formulae for these automorphic forms whenever the underlying Lie group G is split (or complex) and simply laced. Specifically, we review and construct explicitly the minimal representation of G, generalizing the Schrödinger representation of symplectic groups. We compute the spherical vector in this representation, i.e. the wave function invariant under the maximal compact subgroup, which plays the rôle of the summand in the automorphic theta series. We also determine the spherical vector over the complex field. We outline how the spherical vector over the p-adic number fields provides the summation measure in the theta series, postponing its determination to a sequel of this work. The simplicity of our result is suggestive of a new Born–Infeld-like description of the membrane where U-duality is realized non-linearly. Our results may also be used in constructing quantum mechanical systems with spectrum generating symmetries.

An application of Lie groups in distributed control networks

Here we introduce a class of linear operators called recursive orthogonal transforms (ROTs) that allow a natural implementation on a distributed control network. We derive conditions under which ROTs can be used to represent SO( n) for n⩾4. We propose a paradigm for distributed feedback control based on plant matrix diagonalization. To find an ROT suitable for this task, we derive a gradient flow on the appropriate underlying Lie group. A numerical example is presented.

Contraction of superintegrable Hamiltonian systems

We investigate the contraction of a class of superintegrable Hamiltonians by implementing the contraction of the underlying Lie groups. We also discuss the behavior of the coordinate systems that separate their equations of motion, the motion constants, as well as the corresponding solutions along such a process.

Geometric Descent Algorithms for Attitude Determination Using the Global Positioning System

Thispaperdescribesasetofnumericalgradient-basedoptimizationalgorithmsforsolvingtheGlobalPositioning System (GPS)-based attitude determination problem. We pose the problem as one of minimizing the function tr(H NH T Qi 2H W) with respect to the rotation matrix H , where N, Q, and W are given 3 £ 3 matrices, and tr(¢ ) denotes thematrix trace. Both the method of steepest descent and Newton’ s method are generalized to the rotation group by taking advantage of its underlying Lie group structure. Analytic solutions to the line search procedure are also derived. Results of numerical experiments for the class of geometric descent algorithms proposed here are presented and compared with those of traditional vector space-based constrained optimization algorithms.

Self-similar solutions in gas dynamics with exponential time dependence

Self-similar gas flow with similarity coordinate ξ=re∓t, exponential in time, is investigated for plane, cylindrical and spherical geometry. The underlying Lie group symmetry is pointed out and also how it is obtained from flow with power-law self-similarity (ξ=r/|t|α) in the limit α→∞. Six distinct types of solutions are derived and plotted in r−t coordinates. They are characterized by steep density gradients. Also, solutions with strong shock boundaries are discussed. A completely analytical solution is presented, related to Sedov’s solution of a strong point explosion.

Hirota's solitons in the affine and the conformal affine Toda models

We use Hirota's method formulated as a recursive scheme to construct a complete set of soliton solutions for the affine Toda field theory based on an arbitrary Lie algebra. Our solutions include a new class of solitons connected with two different types of degeneracies encountered in Hirota's perturbation approach. We also derive an universal mass formula for all Hirota's solutions to the affine Toda model valid for all underlying Lie groups. Embedding of the affine Toda model in the conformal affine Toda model plays a crucial role in this analysis.

Existence and uniqueness of solutions to superdifferential equations

We state and prove the theorem of existence and uniqueness of solutions to ordinary superdifferential equations on supermanifolds. It is shown that any supervector field, X = X 0 + X 1, has a unique integral flow, Г: R 1¦1 x ( M, A M ) → ( M, A M ), satisfying a given initial condition. A necessary and sufficient condition for this integral flow to yield an R 1¦1-action is obtained: the homogeneous components, X 0, and, X 1, of the given field must define a Lie superalgebra of dimension (1, 1). The supergroup structure on R 1¦1, however, has to be specified: there are three non-isomorphic Lie supergroup structures on R 1¦1, all of which have addition as the group operation in the underlying Lie group R . On the other extreme, even if X 0, and X 1 do not close to form a Lie superalgebra, the integral flow of X is uniquely determined and is independent of the Lie supergroup structure imposed on R 1¦1. This fact makes it possible to establish an unambiguous relationship between the algebraic Lie derivative of supergeometric objects (e.g., superforms), and its geometrical definition in terms of integral flows. It is shown by means of examples that if a supergroup structure in R 1¦1 is fixed, some flows obtained from left-invariant supervector fields on Lie supergroups may fail to define an R 1¦1-action of the chosen structure. Finally, necessary and sufficient conditions for the integral flows of two supervector fields to commute are given.

The point explosion with heat conduction

The influence of nonlinear heat conduction is investigated for strong point explosions in an ambient gas. An ideal gas equation of state and a heat conductivity depending on temperature and density in power-law form are assumed. It is shown that two spherical waves are obtained—a shock wave and a heat wave. They have sharp fronts which run at different speeds, in general, and in a relative order depending on parameters and time. Starting from the underlying Lie group symmetry, self-similar solutions of the problem are discussed in detail; they exist under the assumption that the ambient gas density decays with a given power of the radius. The non-self-similar situation, occurring for uniform density, is also considered. In this case, the shock front first runs behind the heat front, but then overtakes it at a certain time t1. For t≫t1, the well-known hydrodynamic solution of the problem without heat conduction becomes valid, except for a central, almost isobaric region where heat conduction modifies the classical result and keeps the temperature finite. It is shown that this central zone still has the form of a heat wave with a sharp front and evolves self-similarly, though with a smaller similarity exponent than the global hydrodynamic wave. Analytic results for the central temperature and the radial extension of the conduction dominated zone are given.

Anisotropic asymptotic behavior in chaotic inflation

We show that homogeneous cosmological models which are coupled to a scalar field can only approach isotropy at infinite times if the underlying Lie group is admitted by a Friedmann-Robertson-Walker model. This result generalises a theorem of Collins and Hawking for matter satisfying the positive pressure condition to scalar fields with an arbitrary positive, convex potential having a local minimum at φ 0 with V[ φ 0]=0.

Hamiltonian dynamics of a rigid body in a central gravitational field

This paper concerns the dynamics of a rigid body of finite extent moving under the influence of a central gravitational field. A principal motivation behind this paper is to reveal the hamiltonian structure of the n-body problem for masses of finite extent and to understand the approximation inherent to modeling the system as the motion of point masses. To this end, explicit account is taken of effects arising because of the finite extent of the moving body. In the spirit of Arnold and Smale, exact models of spin-orbit coupling are formulated, with particular attention given to the underlying Lie group framework. Hamiltonian structures associated with such models are carefully constructed and shown to benon-canonical. Special motions, namely relative equilibria, are investigated in detail and the notion of anon-great circle relative equilibrium is introduced. Non-great circle motions cannot arise in the point mass model. In our analysis, a variational characterization of relative equilibria is found to be very useful.

Torsion and geometrostasis in nonlinear sigma models

We discuss some general effects produced by adding Wess-Zumino terms to the actions of nonlinear sigma models, an addition which may be made if the underlying field manifold has appropriate homological properties. We emphasize the geometrical aspects of such models, especially the role played by torsion on the field manifold. For general chiral models, we show explicitly that the torsion is simply the structure constant of the underlying Lie group, converted by vielbeine into an antisymmetric rank-three tensor acting on the field manifold. We also investigate in two dimensions the supersymmetric extensions on nonlinear sigma models with torsion, showing how the purely results carry over completely. We consider in some detail the renormalization effects produced by the Wess-Zumino terms using the background field method. In particular, we demonstrate to two-loop order the existence of geometrostasis, i.e. fixed points in the renormalized geometry of the field manifold due to parallelism.