on witten s proof of the positive energy theorem this paper gives a mathematically rigorous proof of the positive energy theorem using spinors this completes and simplifies the original argument presented by edward witten we clarify the geometric aspects of this argument and prove the necessary analytic theorems concerning the relevant dirac operator

degenerations of filippov algebras we consider the variety of filippov n lie algebra structures on an n 1 dimensional vector space the group gln k acts on it and we study the orbit closures with respect to the zariski topology this leads to the definition of filippov algebra degenerations we present some fundamental results on such degenerations including trace invariants and necessary degeneration criteria finally we classify all orbit closures in the variety of complex n 1 dimensional filippov n ary algebras we consider the variety of filippov n lie algebra structures on an n 1 dimensional vector space the group gln k acts on it and we study the orbit closures with respect to the zariski topology this leads to the definition of filippov algebra degenerations we present some fundamental results on such degenerations including trace invariants and necessary degeneration criteria finally we classify all orbit closures in the variety of complex n 1 dimensional filippov n ary algebras

contractions nijenhuis and saletan tensors for general algebraic structures we study generalizations in many directions of the contraction procedure for lie algebras introduced by saletan we consider products of an arbitrary nature not necessarily lie brackets and we generalize to infinite dimension considering a modification of the approach by nijenhuis tensors to bilinear operations on sections of finite dimensional vector bundles we apply our general procedure to lie algebras lie algebroids and poisson brackets we also present results on contractions of n ary products and coproducts

grading refinements in the contractions of lie algebras and their invariants all mutually inequivalent toroidal gradings of the lie algebra are constructed each of them provides several different graded contractions to other eight dimensional lie algebras these can be either continuous or discrete the continuous ones are related to generalized wigner inonu contractions the behaviour of casimir operators under the graded contractions is studied and the casimir and generalized casimir operators of the contracted lie algebras are presented

on lie gradings ii abstract to obtain fine gradings of gl n c it is necessary to study maximal abelian subgroups mad subgroups of diagonizable automorphisms in a ut gl n c we described two finite sets h n and k n such that each mad subgroup consisting only from inner automorphisms is conjugated to some element of h n and each mad subgroup containing at least one outer automorphism is conjugated to some element of k n using these results concerning k n we then easily find all mad subgroups in a ut o n c for n 8 and in a ut sp 2n c

deformations and contractions of lie algebras in this contributed presentation we discuss and compare the mutually opposite procedures of deformations and contractions of lie algebras we suggest that with approp riate combinations of both procedures one may construct new lie algebras we first discuss low dimensional lie algebras and illustrate thereby that whereas for every contraction there exists a reverse deformation the converse is not true in general also we note that some lie algebras belonging to parameterized families are singled out by the irreversibility of defor mations and contractions after reminding that global deformations of the witt virasoro and affine kac moody algebras allow one to retrieve lie algebras of krichever novikov type we contract the latter to find new infinite dimensional lie algebras

unchanged direction motion in general relativity the problems of prescribing acceleration and the extensibility of trajectories the notion of unchanged direction ud motion in general relativity is introduced extending widely the concept of uniformly accelerated motion an observer which obeys an ud motion is characterized as a pointing future unit timelike curve with all its curvatures identically zero up to the first one the initial value problem when the acceleration of the motion is prescribed is analysed it is also studied that the completeness of inextensible ud motions that can be physically interpreted saying that the observers which obey an ud motion lives forever for certain spacetimes with relevant symmetries that includes the generalized robertson walker spacetimes a geometric approach leads to the completeness on the other hand a more analytical approach permits to prove completeness of inextensible ud motions in a plane wave spacetime

contractions of lie algebras and separation of variables the wigner inonu contraction from the rotation group o 3 to the euclidean group e 2 is used to relate the separation of variables in the laplace beltrami operators on two corresponding homogeneous spaces different realizations of the contraction take the two separable coordinate systems on the sphere to the four on the plane

graded contractions of representations of orthogonal and symplectic lie algebras with respect to their maximal parabolic subalgebras parabolic gradings of the classical simple lie algebras o n c n or 5 and sp 2n c n or 2 with complex parameters are described for all maximal parabolic subalgebras all contractions which leave a maximal parabolic subalgebra intact and which preserve a parabolic grading parabolic contractions of lie algebras are found contractions of the irreducible representations for each parabolic contraction of the lie algebra are the main results of the article

on the relative slice problem and four dimensional topological surgery a central open problem in the classification theory of topological fourmanifolds is to determine the validity of four dimensional surgery and fivedimensional s cobordism theorems without fundamental group restrictions by work of m freedman 2 3 the class of groups for which these theorems hold good groups includes the groups of polynomial growth recently freedman and teichner 6 showed that more generally the groups of subexponential growth are good it is expected 3 that the theorems fail for free non abelian groups this conjecture is known as the a b slice problem a more precise conjecture states that the whitehead double of the borromean rings wh bor is not freely slice in this paper we study the relative slice reformulation of this problem introduced in 4 our main theorem may be viewed as a result in link theory while providing some evidence towards the conjecture recall the following definition from 4

graded contractions of casimir operators we describe graded contractions of casimir operators of lie algebras the formalism applies to any casimir operator given as a symmetric form we deal with the quadratic casimir operator in exhaustive detail and indicate the modifications for operators of higher degree

equivalence of diagonal contractions to generalized iw contractions with integer exponents abstract we present a simple and rigorous proof of the claim by weimar woods e weimar woods contractions generalized inonu wigner contractions and deformations of finite dimensional lie algebras rev math phys 12 2000 1505 1529 that any diagonal contraction e g a generalized inonu wigner contraction is equivalent to a generalized inonu wigner contraction with integer parameter powers

lie algebra of anomalously scaled fluctuations for quantum lattice systems it is proven that anomalously scaled fluctuations have a natural lie algebra structure the harmonic lattice in the ground state is given as an illustration of the general theorem

non solvable contractions of semisimple lie algebras in low dimension the problem of non solvable contractions of lie algebras is analysed by means of a stability theorem the problem is shown to be deeply related to the embeddings among semisimple lie algebras and the resulting branching rules for representations with this procedure we determine all deformations of indecomposable lie algebras having a nontrivial levi decomposition onto semisimple lie algebras of dimension n 8 and obtain the non solvable contractions of the latter class of algebras

contractions of low dimensional lie algebras theoretical background of continuous contractions of finite dimensional lie algebras is rigorously formulated and developed in particular known necessary criteria of contractions are collected and new criteria are proposed a number of requisite invariant and semi invariant quantities are calculated for wide classes of lie algebras including all low dimensional lie algebras an algorithm that allows one to handle one parametric contractions is presented and applied to low dimensional lie algebras as a result all one parametric continuous contractions for both the complex and real lie algebras of dimensions not greater than 4 are constructed with intensive usage of necessary criteria of contractions and with studying correspondence between real and complex cases levels and colevels of low dimensional lie algebras are discussed in detail properties of multiparametric and repeated contractions are also investigated

r matrix lax pair and multiparameter decompositions of lie algebras we construct r operators on the lie algebra x1d524 x1d529 n or more generally hilbert schmidt operators l2 in hilbert space these operators are related to a multiparameter deformation given by a sequence of parameters a1 a2 we determine for which choices of parameters r operators are r matrices we also construct the lax pair for the corresponding hamilton equations

fluid space times and conharmonic symmetries the conharmonic curvature tensor is considered as an invariant of the conharmonic transformation defined by ishii and the necessary and sufficient conditions for the conharmonic curvature tensor in a perfect fluid space time to be divergence free has been obtained conharmonic motion conharmonic collineation and conharmonic curvature collineation are introduced as subcases of conformal motion conformal collineation and weyl conformal collineation respectively and relations of conharmonic symmetries with inheriting symmetries are investigated in the case of an existing conharmonic killing vector along the flow vector along the anisotropy vector and perpendicular to both in an anisotropic fluid space time it is found that no equation of state is singled out unless the conharmonic killing vector is also a curvature inheritance vector conditions are obtained for the symmetries of the anisotropic fluid space time admitting a conharmonic killing vector to be inherited in the case of a conharmonic sym

classical and quantum superintegrability with applications a superintegrable system is roughly speaking a system that allows more integrals of motion than degrees of freedom this review is devoted to finite dimensional classical and quantum superintegrable systems with scalar potentials and integrals of motion that are polynomials in the momenta we present a classification of second order superintegrable systems in two dimensional riemannian and pseudo riemannian spaces it is based on the study of the quadratic algebras of the integrals of motion and on the equivalence of different systems under coupling constant metamorphosis the determining equations for the existence of integrals of motion of arbitrary order in real euclidean space e2 are presented and partially solved for the case of third order integrals a systematic exposition is given of systems in two and higher dimensional space that allow integrals of arbitrary order the algebras of integrals of motions are not necessarily quadratic but close polynomially or rationally the relation between superintegrability and the classification of orthogonal polynomials is analyzed

the three dimensional real lie algebras and their contractions in this paper the general contractions of the three dimensional real lie algebras are determined a short summary will be given of the history definitions properties examples and applications of contractions of lie algebras so far mostly simple or generalized inonu wigner contractions have been used the three dimensional real lie algebras are classified in such a way that their general contractions can be easily read off all these contractions can be realized by generalized inonu wigner contractions which supports the argument that these are the interesting objects to study

2 1 kinematical expansions from galilei to de sitter algebras expansion of a lie algebra is the opposite process to contraction starting from a lie algebra the expansion process goes to another algebra which is non isomorphic and less abelian we propose an expansion method based on the casimir invariants of the initial and expanded algebras and where the free parameters involved in the expansion are the curvatures of their associated homogeneous spaces this method is applied for expansions within the family of lie algebras of three dimensional spaces and 2 1 d kinematical algebras we show that these expansions are classed into two types the first type makes the curvature of space or spacetime different from zero i e it introduces a space or universe radius while the other has a similar interpretation for the curvature of the space of worldlines which is non positive and equal to 1 c2 in the kinematical algebras we obtain expansions which go from galilei to either newton hooke or poincarealgebras and from these to de sitter algebras as well as some other examples

discrete and continuous graded contractions of lie algebras and superalgebras grading preserving contractions of lie algebras and superalgebras of any type over the complex number field are defined and studied such contractions fall naturally into two classes the wigner inonu like continuous contractions and new discrete contractions a general method is described for any abelian grading semigroup and any lie algebra or superalgebra admitting such a grading all contractions preserving z2 z3 and z2 z2 gradings are found examples of these gradings and contractions for the simple lie algebra a2 affine kac moody algebra a 1 and the simple superalgebra osp 2 1 are shown

graded contractions of the gell mann graded sl 3 c the gell mann grading one of the four gradings of sl 3 c that cannot be further refined is considered as the initial grading for the graded contraction procedure using the symmetries of the gell mann grading the system of contraction equations is reduced and solved each non trivial solution of this system determines a lie algebra which is not isomorphic to the original algebra sl 3 c the resulting 53 contracted algebras are divided into two classes the first is represented by the algebras which are also continuous inonu wigner contractions the second is formed by the discrete graded contractions

casson towers and slice links we prove that a casson tower of height 4 contains a flat embedded disc bounded by the attaching circle and we prove disc embedding results for height 2 and 3 casson towers which are embedded into a 4 manifold with some additional fundamental group assumptions in the proofs we create a capped grope from a casson tower and use a refined height raising argument to establish the existence of a symmetric grope which has two layers of caps data which is sufficient for a topological disc to exist with the desired boundary as applications we present new slice knots and links by giving direct applications of the disc embedding theorem to produce slice discs without first constructing a complementary 4 manifold in particular we construct a family of slice knots which are potential counterexamples to the homotopy ribbon slice conjecture

lie algebra contractions and symmetries of the toda hierarchy the lie algebra l of generalized and point symmetries of the equations in the toda hierarchy is shown to be a semidirect sum of two infinite dimensional lie algebras one perfect the other abelian in the continuous limit the structure of the lie algebra changes a contraction occurs with the lattice spacing as the contraction parameter in particular for the toda equation itself a set of five elements involving both point symmetries and generalized ones contracts to the point symmetry algebra of the potential kdv equation

graded contractions of lie algebras and central extensions we use the concept of grading of lie algebras to investigate the appearance of central charges during the contraction process as for the usual graded contractions one finds simultaneously the central extensions of classes of algebras rather than specific lie algebras to illustrate the method we consider in detail two physical applications the kinematical algebras of spacetime and the u n bosons limits of some lie algebras

subjoinings of affine kac moody algebras subjoinings among affine kac moody algebras is a relation generalising the algebra subalgebra relation it is analogous to subjoinings among semisimple finite lie algebras a list of maximal subjoinings of equal rank affine kac moody algebras is presented it is conjectured that there are no other maximal subjoinings

special conformal killing vector space times and symmetry inheritance viscous heat conducting fluid and anisotropic fluid space times admitting a special conformal killing vector sckv are studied and some general theorems concerning the inheritance of the symmetry associated with the sckv are proved in particular for viscous fluid space times it is shown that i if the sckv maps fluid flow lines into fluid flow lines then all physical components of the energy momentum tensor inherit the sckv symmetry or ii if the lie derivative along a sckv of the shear viscosity term ab is zero then again we have symmetry inheritance all space times admitting a sckv and satisfying the dominant energy condition are found apart from the vacuum pp wave solutions which are the only vacuum solutions that can admit a sckv the energy momentum tensor associated with these space times is shown to admit at least one null eigenvector and can represent either a viscous fluid with heat conduction or an anisotropic fluid no perfect fluid space times can admit a sckv these sckv space t

spatial and lorentzian surfaces in robertson walker space times let l14 f c i fs gfc be a robertson walker space time which does not contain any open subset of constant curvature in this paper we provide a general study of nondegenerate surfaces in l14 f c first we prove the nonexistence of marginally trapped surfaces with positive relative nullity then we classify totally geodesic submanifolds finally we classify the family of surfaces with parallel second fundamental form and the family of totally umbilical surfaces with parallel mean curvature vector

bianchi spaces and their three dimensional isometries as s expansions of two dimensional isometries in this paper we show that certain three dimensional isometry algebras specifically those of type i ii iii and v according to bianchi s classification can be obtained as expansions of the isometries in two dimensions in particular we use the so called s expansion method which makes use of the finite abelian semigroups because it is the most general procedure known until now also it is explicitly shown why it is impossible to obtain the algebras of type iv vi ix as expansions from the isometry algebras in two dimensions all the results are checked with computer programs this procedure shows that the problem of how to relate by an expansion two lie algebras of different dimensions can be entirely solved in particular the procedure can be generalized to higher dimensions which could be useful for diverse physical applications as we discuss in our conclusions

the construction of solvable lie algebras from equidimensional nilpotent algebras abstract the study of gradings of solvable lie algebras l of finite dimensionover a field f of zero characteristic led the authors to the discovery of equidimensional nilpotent algebras l uniquely determined by l up to f isomorphy conversely for any finite dimensional lie algebra l over f an algorithm is developed which yields in parametric form all solvable lie algebras l determining l as the corresponding nilpotent algebra the exposition is independent of lie grading theory it organizes in a novel way the classification of solvable lie algebras of given dimension around the same task for nilpotent algebras every isomorphy class of solvable f algebras is obtained in this way

the four sets of additive quantum numbers of su 3 the four maximal sets of additive quantum numbers and related fine gradings of the lie algebra sl 3 c are described in detail the quantum numbers are determined by a grading of the lie algebra fine gradings providing maximal sets of them two sets of additive quantum numbers are equivalent precisely if the corresponding gradings are equivalent under a transformation from the automorphism group of the lie algebra

4 manifold topology i subexponential groups the technical lemma underlying the 5 dimensional topologicals cobordism conjecture and the 4 dimensional topological surgery conjecture is a purely smooth category statement about locating 1 null immersions of disks these conjectures are theorems precisely for those fundamental groups good groups where the 1 null disk lemma ndl holds we expand the class of known good groups to all groups of subexponential growth and those that can be formed from these by a finite number of application of two operations 1 extension and 2 direct limit the finitely generated groups in this class are amenable and no amenable group is known to lie outside this class

solvable lie algebras of dimension 4 over perfect fields abstract the solvable lie algebras of dimension not greater than four over a perfect field of reference are described in terms of their nilpotent frame counted over finite fields of q elements and classified over the real and complex number fields the number dq n of nonisomorphic solvable lie algebras of dimension n over f q roughly speaking grows as a polynomial in n with qn 2 as highest term but there are smaller additional terms depending on the residue class of q modulo n 1 if n 4

geometrical aspects of the lie algebra s expansion procedure in this article it is shown that s expansion procedure affects the geometry of a lie group changing it and leading us to the geometry of another lie group with higher dimensionality a method for determining the semigroup which would provide a lie algebra from another is outlined via an example finally it is proved that a lie algebra obtained from another lie algebra via s expansion is a non simple lie algebra

graded contractions of representations of special linear lie algebras with respect to their maximal parabolic subalgebras parabolic gradings of the classical simple lie algebras sl n c n or 3 are described for all maximal parabolic subalgebras all contractions are found which leave a maximal parabolic subalgebra intact and which preserve a parabolic grading parabolic contractions of lie algebras contractions of the irreducible representations of sl n c for each parabolic contraction of the lie algebra are the main results of the article

contractions of lie algebras and the separation of variables interbase expansions lie algebra contractions from o n 1 to e n are used to obtain asymptotic limits of interbase expansions between bases corresponding to different subgroup chains for the group o n 1 the contractions lead to interbase expansions for different subgroup chains of the euclidean group e n they provide asymptotic formulae for quantities such as wigner rotation matrices clebsch gordan coefficients and racah coefficients

the general solution of the real graded contractions of the general solution of the graded contraction equations for a grading of the real compact simple lie algebra so n 1 is presented in an explicit way it turns out to depend on independent real parameters the structure of the general graded contractions is displayed for the low dimensional cases and kinematical algebras are shown to appear straightforwardly the geometrical or physical meaning of the contraction parameters as curvatures is also analysed in particular for kinematical algebras these curvatures are directly linked to geometrical properties of possible homogeneous spacetimes

contraction of lie algebra representations motivated by an example of inonu and wigner a general procedure for the contraction of irreducible unitary representations of semisimple compact lie algebras is given in this situation contractions mean generalized inonu wigner contractions in particular this means that by a simple limiting process one obtains representations of nonsemisimple noncompact lie algebras as examples a general treatment of so 3 is given as well as some results for su 3

on the identification of a lie algebra given by its structure constants i direct decompositions levi decompositions and nilradicals abstract methods are given for identifying a lie algebra l given by its structure constants the identification involves a transformation to a canonical basis in which the structure becomes obvious thus decomposable lie algebras are already decomposed into direct sums of indecomposable ones an indecomposable lie algebra that is not simple or solvable has its radical exhibited a solvable algebra has its nilradical displayed the methods all lead to simple algorithms that have been implemented as computer programs involving some symbolic manipulations

new gradings of sl 3 c representations two new gradings of the lie algebra sl 3 c are the refined sl 2 c and o 3 gradings the grading in each case utilizes the sl 2 c or o 3 weight together with a new additive modular label a complete set of sl 3 c representation basis states labeled by each to the two sets of additive quantum numbers is found the newlabeling operator in both cases fails to commute with the cubic sl 3 c casimir operator and hence mixes states of the contragredient representations p q and q p

asymptotically flat space times have no conformal killing fields it is shown that asymptotically flat space times with certain properties do not admit conformal killing fields which are not killing fields the space times must be vacuum asymptotically minkowskian and have positive bondi energy

lowest dimensional example on non universality of generalized inonu wigner contractions abstract we prove that there exists just one pair of complex four dimensional lie algebras such that a well defined contraction among them is not equivalent to a generalized iw contraction or to a one parametric subgroup degeneration in conventional algebraic terms over the field of real numbers this pair of algebras is split into two pairs with the same contracted algebra the example we constructed demonstrates that even in the dimension four generalized iw contractions are not sufficient for realizing all possible contractions and this is the lowest dimension in which generalized iw contractions are not universal moreover this is also the first example of nonexistence of generalized iw contraction for the case when the contracted algebra is not characteristically nilpotent and therefore admits nontrivial diagonal derivations the lower bound equal to three of nonnegative integer parameter exponents which are sufficient to realize all generalized iw contractions of four dimensional lie algebras is also found

central extensions of the quasi orthogonal lie algebras we determine the central extensions of a whole family of lie algebras obtained by the method of graded contractions from n arbitrary all the inhomogeneous orthogonal and pseudo orthogonal algebras are members of this family as well as a large number of other non semisimple algebras all of which have at least a semidirect structure in some cases two or more the dimensions of their second cohomology groups and the explicit expression of their central extensions are given

how solvable is 2 1 dimensional einstein gravity in this paper the relationship between witten s approach to the 2 1 dimensional vacuum einstein equations for spatially compact space times and the conventional annowitt deser and misner adm hamiltonian approach is discussed it is argued especially for the space times with higher genus cauchy surfaces that neither approach is complete in itself witten s because it does not provide a technique even at the classical level for recovering the space time metric and the conventional approach because it provides no mechanism for solving a seemingly intractable set of hamilton equations it is also argued however that the two formulations are instead complementary in the sense that the wilson loops which play a key role in witten s approach provide at least in principle a mechanism for solving the reduced hamilton equations and thereby completing the picture at the classical level an example of this synthesis for the explicitly computable case of genus one hypersurfaces is provided the m

graded contractions of the lie algebra e 2 1 the graded contractions of pseudo euclidean lie algebra are studied the non equivalent gradings of of type and a are extended to the entire lie algebra using the action of on the abelian ideal the translations the graded contractions embed into a large family of six dimensional lie algebras the family includes solvable nilpotent and nonsolvable lie algebras both decomposable and indecomposable ones the distinction between graded and inonu wigner contractions is analysed the physically most interesting lie algebras obtained by the contractions are the inhomogeneous galilei and pseudo galilei

graded contractions of sl 3 c the wigner inonu contraction is generalized to include gradings other than z2 and the formalism is applied to obtain all toroidal contractions of sl 3 c examples of contractions based on nontoroidal gradings are also given

proof of the positive mass theorem ii the positive mass theorem states that for a nontrivial isolated physical system the total energy which includes contributions from both matter and gravitation is positive this assertion was demonstrated in our previous paper in the important case when the space time admits a maximal slice here this assumption is removed and the general theorem is demonstrated abstracts of the results of this paper appeared in 11 and 13

quantum structure of the motion groups of the two dimensional cayley klein geometries a simultaneous and global scheme of quantum deformation is defined for the set of algebras corresponding to the groups of motions of the two dimensional cayley klein geometries their central extensions are also considered under this unified pattern in both cases some fundamental properties characterizing the classical ck geometries as the existence of a set of commuting involutions contractions and dualities relationships remain in the quantum version

graded contractions and kinematical groups of space time g graded contractions of the complex lie algebra b2 and of its real forms are considered it is found that a particular g z2 z2 is a proper choice of the grading group preserved when graded contractions leading to lie algebras of the kinematical groups of space time are studied

on lie gradings iii gradings of the real forms of classical lie algebras abstract maximal abelian subgroups of diagonalizable automorphisms of lie algebra so called mad groups play a crucial role in the construction of fine gradings of lie algebra our aim is to give a description of mad groups for real forms of classical lie algebras we introduce four types of matrix subgroups of g l n c called out groups ad groups out groups and ad groups for each type of these subgroups we define a relation of equivalence the problem of classifying of all non conjugate mad groups on real forms of sl n c o n c or sp n c is transformed to the problem of classifying these equivalence classes the classification of these equivalence classes is presented here

conformal killing initial data we find necessary and sufficient conditions ensuring that the vacuum development of an initial dataset of einstein s field equations admits a conformal killing vector we refer to these conditions as conformal killing initial data ckid and they extend the well known killing initial data kid that have been known for a long time the procedure used to find the ckid is a classical argument which is reviewed and presented in a form that may have an independent interest based on identifying a suitable propagation identity and checking the well posedness of the corresponding initial value problem as example applications we review the derivation of the kid conditions as well as give a more thorough treatment of the homothetic kid conditions than was previously available in the literature we find necessary and sufficient conditions ensuring that the vacuum development of an initial dataset of einstein s field equations admits a conformal killing vector we refer to these conditions as conformal killing initial data ckid and they extend the well known killing initial data kid that have been known for a long time the procedure used to find the ckid is a classical argument which is reviewed and presented in a form that may have an independent interest based on identifying a suitable propagation identity and checking the well posedness of the corresponding initial value problem as example applications we review the derivation of the kid conditions as well as give a more thorough treatment of the homothetic kid conditions than was previously available in the literature

solutions of einstein s equations with conformal symmetries and a contact slice abstract first we show that the initial space like slice of the generalized robertson walker grw space time is einstein if and only if the electric part of the space time weyl conformal tensor vanishes we also show that the purely spatial component of the space time weyl tensor vanishes if and only if the initial slice has constant curvature then assuming that a spatially complete synchronous space time solution of einstein s equations admits a conformal vector field whose normal component is non constant and spatial component is conformal on each space like slice we show that each slice is conformal to i a euclidean space ii a sphere or iii a hyperbolic space or iv riemannian product of an open interval and and a riemannian space finally under the preceding hypotheses if a slice has a k contact metric then it is isometric to a unit sphere

contractions of lie algebras generalized inonu wigner contractions versus graded contractions recently a new notion of contraction has been introduced the purely algebraically defined graded contractions that fall into two disjoint classes continuous and discrete our main result is that all continuous graded contractions can be realized by generalized inonu wigner contractions furthermore we give a complete characterization of the discrete graded contractions via certain higher order identities inherent to the grading group

proper conformal symmetries of space times with divergence free weyl conformal tensor it has been shown that a four dimensional space time with divergence free weyl conformal curvature tensor and admitting a proper conformal symmetry is locally of type o conformally flat or n the latter case has been studied in terms of null sectional curvature finally a physical characterization of general relativistic perfect fluid space times with divergence free weyl conformal tensor has been presented and moreover it is shown that such fluid space times admitting a proper conformal symmetry are conformally flat

conformal symmetries of einstein s field equations and initial data this paper examines the initial data for the evolution of the space time solution of einstein s equations admitting a conformal symmetry under certain conditions on the extrinsic curvature of the initial complete spacelike hypersurface and sectional curvature of the space time with respect to sections containing the normal vector field we have shown that the initial hypersurface is conformally diffeomorphic to a sphere or a flat space or a hyperbolic space or the product of an open real interval and a complete 2 manifold it has been further shown that if the initial hypersurface is compact then it is conformally diffeomorphic to a sphere finally the conformal symmetries of a generalized robertson walker space time have been described

cayley klein algebras as graded contractions of so n 1 we study z2 x n graded contractions of the real compact simple lie algebra so n 1 and we identify within them the cayley klein algebras as a naturally distinguished subset

models of the lie algebra f4 abstract we describe the models of the exceptional lie algebra f 4 which are based on its semisimple subalgebras of rank 4 the underlying fact is that any reductive subalgebra of maximal rank of a simple lie algebra induces a grading on this algebra by means of an abelian group in such a way that the nontrivial components of the grading are irreducible modules

a counterexample to the strong version of freedman s conjecture a long standing conjecture due to michael freedman asserts that the 4 dimensional topological surgery conjecture fails for non abelian free groups or equivalently that a family of canonical examples of links the generalized borromean rings are not a b slice a stronger version of the conjecture that the borromean rings are not even weakly a b slice where one drops the equivariant aspect of the problem has been the main focus in the search for an obstruction to surgery we show that the borromean rings and more generally all links with trivial linking numbers are in fact weakly a b slice this result shows the lack of a non abelian extension of alexander duality in dimension 4 and of an analogue of milnor s theory of link homotopy for general decompositions of the 4 ball

conformal killing vector fields on spacetime solutions of einstein s equations and initial data abstract working within the adm formalism we have established results characterizing a conformal killing vector ckv field on a spacetime solution of einstein s field equations and the initial data on a spacelike hypersurface we have considered the cases i the ckv is tangential to spacelike slices ii the ckv is closed and the initial hypersurface is totally umbilical and has constant mean curvature cmc and iii spacetime is a perfect fluid has a non vanishing ckv transverse to a compact cmc hypersurface orthogonal to the 4 velocity finally we show that a non null ckv that is also a geodesic vector field is either homothetic or locally closed

discrete and continuous graded contractions of representations of lie algebras simultaneous grading of lie algebras and their representation spaces is used to develop a new theory of grading preserving contractions of representations of all lie algebras admitting the chosen grading the theory is completely different from the traditional ways of contracting representations the graded contractions fall naturally into two classes discrete and continuous ones related respectively to 2 cocycles and coboundaries of the grading group

general properties of the expansion methods of lie algebras the study of the relation between lie algebras and groups and especially the derivation of new algebras from them is a problem of great interest in mathematics and physics because finding a new lie group from an already known one also means that a new physical theory can be obtained from a known one one of the procedures that allow us to do so is called expansion of lie algebras and has been recently used in different physical applications particularly in gauge theories of gravity here we report on further developments of this method required to understand in a deeper way their consequences in physical theories we have found theorems related to the preservation of some properties of the algebras under expansions that can be used as criteria and more specifically as necessary conditions to know if two arbitrary lie algebras can be related by some expansion mechanism formal aspects such as the cartan decomposition of the expanded algebras are also discussed finally an instructive example that allows us to check explicitly all our theoretical results is also provided

a berger green type inequality for compact lorentzian manifolds we give a lorentzian metric on the null congruence associated with a timelike conformal vector field a liouville type theorem is proved and a boundedness for the volume of the null congruence analogous to a well known berger green theorem in the riemannian case will be derived by studying conjugate points along null geodesics as a consequence several classification results on certain compact lorentzian manifolds without conjugate points on its null geodesics are obtained finally several properties of null geodesics of a natural lorentzian metric on each odd dimensional sphere have been found

anti de sitter poincare symmetries and representations from poincare galilei through a classical deformation approach a classical deformation procedure based on universal enveloping algebras casimirs and curvatures of symmetrical homogeneous spaces is applied to several cases of physical relevance starting from the 3 1 d galilei algebra we describe at the level of representations the process leading to its two physically meaningful deformed neighbours the poincare algebra is obtained by introducing a negative curvature in the flat galilean phase space or space of worldlines while keeping a flat spacetime to be precise starting from a representation of the galilei algebra with both casimirs different from zero we obtain a representation of the poincare algebra with both casimirs necessarily equal to zero the poincare angular momentum pauli lubanski components position and velocity operators etc are expressed in terms of galilean operators through some expressions deforming the proper galilean ones similarly the newton hooke algebras appear by endowing spacetime with a nonzero curvature while keeping a flat phase space the same approach starting from the 3 1 d poincare algebra provides representations of the anti de sitter as poincare deformations

lie bialgebra contractions and quantum deformations of quasi orthogonal algebras lie bialgebra contractions are introduced and classified a non degenerate coboundary bialgebra structure is implemented into all pseudo orthogonal real algebras so p q starting from the one corresponding to so n 1 it allows us to introduce a set of lie bialgebra contractions which leads to lie bialgebras of quasi orthogonal algebras this construction is explicitly given for n 2 3 4 all lie bialgebra contractions studied in this paper define hopf algebra contractions for the drinfel d jimbo deformations uzso p q they are explicitly used to generate new non semisimple quantum algebras as it is the case for the euclidean poincare and galilean algebras

graded contractions and the conformal group of minkowski space time all z2 z2 graded contractions preserving the space isotropy with grading induced by the space inversion and time reversal automorphisms of the lie algebras a3 and so 4 2 are listed some properties of these contractions are discussed

contractions of lie algebras and separation of variables the n dimensional sphere inonu wigner contractions from the rotation group o n 1 to the euclidean group e n are used to relate the separation of variables in laplace beltrami operators on n dimensional spheres and euclidean spaces respectively in this article we consider all subgroup type coordinates corresponding to different chains of subgroups of o n 1 and e n respectively in particular the contractions relate the graphical formalism of trees on spheres to the clusters on euclidean spaces introduced in this article the contractions are considered analytically on several levels the vector fields realizing the lie algebras the complete sets of commuting operators characterizing separable coordinate systems the coordinate systems themselves and the separated eigenfunctions

homothetic and conformal symmetries of solutions to einstein s equations we present several results about the nonexistence of solutions of einstein s equations with homothetic or conformal symmetry we show that the only spatially compact globally hyperbolic spacetimes admitting a hypersurface of constant mean extrinsic curvature and also admitting an infinitesimal proper homothetic symmetry are everywhere locally flat this assumes that the matter fields either obey certain energy conditions or are the yang mills or massless klein gordon fields we find that the only vacuum solutions admitting an infinitesimal proper conformal symmetry are everywhere locally flat spacetimes and certain plane wave solutions we show that if the dominant energy condition is assumed then minkowski spacetime is the only asymptotically flat solution which has an infinitesimal conformal symmetry that is asymptotic to a dilation in other words with the exceptions cited homothetic or conformal killing fields are in fact killing in spatially compact or asymptotically flat spactimes in the conformal procedure for solving the initial value problem we show that data with infinitesimal conformal symmetry evolves to a spacetime with full isometry