Geometrization of metric boundary data for Einstein’s equations

The principle part of Einstein equations in the harmonic gauge consists of a constrained system of 10 curved space wave equations for the components of the space-time metric. A well-posed initial boundary value problem based upon a new formulation of constraint-preserving boundary conditions of the Sommerfeld type has recently been established for such systems. In this paper these boundary conditions are recast in a geometric form. This serves as a first step toward their application to other metric formulations of Einstein’s equations.     

Impulse formulations of Hall magnetohydrodynamic equations

Impulse formulations of Hall magnetohydrodynamic (MHD) equations are developed. The Lagrange invariance of a generalized ion magnetic helicity is established for Hall MHD. The physical implications of this Lagrange invariant are discussed. The discussion is then extended to compressible Hall MHD and a generalized ion magnetic potential helicity Lagrange invariant is established. The physical implications of the generalized ion magnetic potential helicity Lagrange invariant are shown to be the same, as to be expected, as those of the generalized ion magnetic helicity Lagrange invariant.     

Bilocal lie groups and solitons

A group theoretical scheme where solition equations are associated with the integrability conditions for differential system is proposed. These conditions ensure the existence of a bilocal Lie group structure which naturally generates a set of conserved currents for arbitrary space-time dimensions.     

Nonlinear differential equations and lie superalgebras

It is shown how the theory of nonlinear ordinary differential equations with superposition formulas can be generalized to the case of superequations involving anticommuting Grassmann variables. As an example, equations based on the osp(1, 2) superalgebra are analyzed in detail. They turn out to be super-Riccati equations and a super-superposition formula is obtained for their solutions.

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Résumé Nous montrons comment la théorie des équations différentielles ordinaires non-linéaires admettant des lois de superposition peut être généralisée au cas de super-équations contenant des variables de Grassmann anticommutantes. Comme exemple les équations basées sur la super-algèbre osp(1, 2) sont analysées en détail. Nous obtenons des super-équations de Riccati et leurs formules de superposition sont mises en évidence.

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Chargé de recherches du FNRS.     

Some spherical solutions of ideal magnetohydrodynamic equations

Abstract

Some spherical solutions of the ideal magnetohydrodynamic (MHD) equations are obtained from the method of the weak transversality method (WTM), which is based on Lie group theory. This analytical method makes use of the symmetry group of the MHD system in situations where the “classical” Lie approach of symmetry reductions is no longer applicable. Also, a brief physical interpretation of these solutions is given.     

Differential calculus on quantized simple lie groups

Differential calculi, generalizations of Woronowicz's four-dimensional calculus on SU _q (2), are introduced for quantized classical simple Lie groups in a constructive way. For this purpose, the approach of Faddeev and his collaborators to quantum groups was used. An equivalence of Woronowicz's enveloping algebra generated by the dual space to the left-invariant differential forms and the corresponding quantized universal enveloping algebra, is obtained for our differential calculi. Real forms for q are also discussed.     

Nonlinear representations of connected nilpotent real lie groups

We consider nonlinear representations such that the linear part is the finite sum of irreducible unitary representations, in which case linearizations and canonical forms (normalizations) are given and we prove the analyticity of these normalizations in good cases.     

An interesting representation of lie algebras of linear groups

I have presented a means of getting a representation space of a general linear group ofn dimensions in terms of homogeneous functions ofn,n-dimensional vectors. Except in particular cases, the representation is of the Lie algebra, rather than the group. A general formalism is set up to evaluate the Casimir operators of the Lie algebra of the group in terms of the degrees of homogeneity of the functions (which are eigenfunctions of the Casimir operators) in then variables. It is noticed that the Casimir operators exhibit certain symmetries in these degrees of homogeneity which relate different representations having the same eigenvalues for the Casimir operators. Contour integral formulas that enable one to pass from one such representation to another are presented. An expression for the eigenvalues of a general Casimir operator in terms of the degree of homogeneity is presented.     

Nonautonomous submersive second-order differential equations and lie symmetries

We give necessary and sufficient conditions for a nonautonomous second-order differential equation to be submersive. An application to nonautonomous Lagrangian systems is given: the existence of symmetries of the Lagrangian permits us to prove that the Euler-Lagrange vector field is submersive and hence that the motion equations may be simplified. Our results extend to the nonautonomous case the previous ones obtained by Kossowski and Thompson.     

Geometrization of quantum mechanics

Quantum mechanics is cast into a classical Hamiltonian form in terms of a symplectic structure, not on the Hilbert space of state-vectors but on the more physically relevant infinite-dimensional manifold of instantaneous pure states. This geometrical structure can accommodate generalizations of quantum mechanics, including the nonlinear relativistic models recently proposed. It is shown that any such generalization satisfying a few physically reasonable conditions would reduce to ordinary quantum mechanics for states that are near the vacuum. In particular the origin of complex structure is described.     

Adaptive registration of diffusion tensor images on lie groups

With diffusion tensor imaging (DTI), more exquisite information on tissue microstructure is provided for medical image processing. In this paper, we present a locally adaptive topology preserving method for DTI registration on Lie groups. The method aims to obtain more plausible diffeomorphisms for spatial transformations via accurate approximation for the local tangent space on the Lie group manifold. In order to capture an exact geometric structure of the Lie group, the local linear approximation is efficiently optimized by using the adaptive selection of the local neighborhood sizes on the given set of data points. Furthermore, numerical comparative experiments are conducted on both synthetic data and real DTI data to demonstrate that the proposed method yields a higher degree of topology preservation on a dense deformation tensor field while improving the registration accuracy.     

Analysis of some monomial representations of exponential solvable lie groups

Let G be an exponential solvable Lie group and H an analytic subgroup of G. Let x be a unitary character of H and τ = Ind _H ^G χ. We provide a necessary and a sufficient condition for the representation τ to split finitely on its isotopic components for adapted triplets. For the case in which H is maximal, we provide a concrete and smooth intertwining operator and its inverse for the disintegration of τ into irreducibles. Research supported by the D.G.R.S.T., Unity 00/UR/15-01.     

Van der Waerden continuity theorem for semisimple lie groups

As was proved by van der Waerden in 1933, every finite-dimensional locally bounded representation of a semisimple compact Lie group is continuous. This is the famous “van der Waerden continuity theorem,” and it motivated a vast literature. In particular, relationships between the assertion of the theorem (and of the inverse, in a sense, to this theorem) and some properties of the Bohr compactifications of topological groups were established, which led to the introduction and the study of certain classes of the so-called van der Waerden groups and algebras. Until now, after more than 70 years have passed, the van der Waerden theorem appears in monographs and surveys in diverse forms; new proofs were found and then simplified in important special cases. In this paper, we prove that the statement of the van der Waerden theorem remains valid for all (not necessarily compact) real semisimple Lie groups, i.e., any given finite-dimensional representation of a real semisimple Lie group is continuous if and only if this representation is locally bounded. More subtle results are also obtained. The main theorem contains several conditions equivalent to the continuity condition for a finite-dimensional representation. In particular, it is proved that every finite-dimensional representation of a real semisimple Lie group is continuous if and only if the restriction of this representation to the “compact” part, to the Abelian part, or to the nilpotent part of the Iwasawa decomposition is locally bounded, and the original van der Waerden theorem is also somewhat refined. For instance, the following assertion holds: every finite-dimensional representation of a compact semisimple Lie group is continuous if and only if the restriction of this representation to some maximal torus is locally bounded. Dedicated to the memory of George Mackey (1916–2006) Partially supported by the Russian Foundation for Basic Research under grant no. 02-01-00574, by the Program of Supporting the Leading Scientific Schools under grant no. NSh 619.203.1, and by the INTAS grant.     

Vacuum polarization of a scalar field on the nonunimodular lie groups

A method for calculating the vacuum average energy-momentum tensors for a scalar field on the nonunimodular Lie groups is developed. To solve this problem, a method of generalized harmonic analysis on the Lie groups is used based on the method of orbits of coadjoint representation.     

Superposition rules, lie theorem, and partial differential equations

A rigorous geometric proof of the Lie theorem on nonlinear superposition rules for solutions of nonautonomous ordinary differential equations is given filling in all the gaps present in the existing literature. The proof is based on an alternative but equivalent definition of a superposition rule: it is considered as a foliation with some suitable properties. The problem of uniqueness of the superposition function is solved, the key point being the codimension of the foliation constructed from the given Lie algebra of vector fields. Finally, as a more convincing argument supporting the use of this alternative definition of superposition rule, it is shown that this definition allows an immediate generalization of the Lie theorem for the case of systems of partial differential equations.     

Von Neumann kernels of connected lie groups,revisited and refined

In the note, a 30-year old result concerning the von Neumann kernel of a connected Lie group (the intersection of kernels of all irreducible finite-dimensional continuous complex unitary representations of the group) is corrected and the smallest von Neumann kernel of a connected Lie group (the intersection of kernels of all irreducible finite-dimensional (not necessarily continuous) complex unitary representations of the group) is described.     

Geometrization of String Theory Gravity

It is shown that the low energy string theory Lagrangian can be interpreted as pure gravity. In particular, it is shown that the Lagrangian is simply R, the curvature scalar of spacetime with torsion, and unlike in previous work, the covariant derivative of the metric tensor vanishes. As a consequence, it is shown that the physical origin of the scalar field results from the pseudoscalar invariant. This yields, for the first time, a definite physical origin of the dilaton field in four dimensions.     

Solvability of the inhomogeneous mixed boundary value problem for stationary magnetohydrodynamic equations

Global solvability of the boundary value problem for stationary magnetohydrodynamic equations considered under inhomogeneous mixed boundary conditions for a magnetic field and the inhomogeneous Dirichlet condition for the velocity is proved.