superalgebra realization of the 3 algebras in n 6 8 chern simons matter theories we use superalgebras to realize the 3 algebras used to construct n 6 8 chern simons matter csm theories we demonstrate that the superalgebra realization of the 3 algebras provides a unified framework for classifying the gauge groups of the n 5 theories based on 3 algebras using this realization we rederive the ordinary lie algebra construction of the general n 6 csm theory from its 3 algebra counterpart and reproduce all known examples as well in particular we explicitly construct the nambu 3 bracket in terms of a double graded commutator of psu 2 2 the n 8 theory of bagger lambert and gustavsson blg with so 4 gauge group is constructed by using several different ways a quantization scheme for the 3 brackets is proposed by promoting the double graded commutators as quantum mechanical double graded commutators

algebraic vs physical n 6 3 algebras in our previous paper we classified linearly compact algebraic simple n 6 3 algebras in the present paper we classify their physical counterparts which actually appear in the n 6 supersymmetric 3 dimensional chern simons theories

n ary algebras a review with applications 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self contained and unified way these generalizations are algebraic structures in which the two entries lie bracket has been replaced by a bracket with n entries each type of n ary bracket satisfies a specific characteristic identity which plays the role of the jacobi identity for lie algebras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 three lie algebras have surfaced recently in multi brane theory in the context of the bagger lambert gustavsson model because of this 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 a4 model may be formulated in terms of an ordinary lie algebra and on its nambu bracket generalization

hypermatrix factors for string and membrane junctions the adjoint representations of the lie algebras of the classical groups su n so n and sp n are respectively tensor antisymmetric and symmetric products of two vector spaces and hence are matrix representations we consider the analogous products of three vector spaces and study when they appear as summands in lie algebra decompositions the grading of the exceptional lie algebras provides such summands and provides representations of classical groups on hypermatrices the main natural application is a formal study of 3 junctions of strings and membranes generalizations are also considered

triangulations of subanalytic sets and locally subanalytic manifolds if two polyhedrons are locally subanalytically homeomorphic that is the graph is locally subanalytic they are pl homeomorphic a lo cally subanalytic manifold is one whose coordinate transformations are locally subanalytic it is proved that a locally subanalytic manifold has a unique pl manifold structure a semialgebraic manifold also is considered

deligne products of line bundles over moduli spaces of curves we study deligne products for forgetful maps between moduli spaces of marked curves by offering a closed formula for tautological line bundles associated to marked points in particular we show that the deligne products for line bundles on the total spaces corresponding to forgotten marked points are positive integral multiples of the weil petersson bundles on the base moduli spaces

the anomaly line bundle of the self dual field theory in this work we determine explicitly the anomaly line bundle of the abelian self dual field theory over the space of metrics modulo diffeomorphisms including its torsion part inspired by the work of belov and moore we propose a non covariant action principle for a pair of euclidean self dual fields on a generic oriented riemannian manifold the corresponding path integral allows one to study the global properties of the partition function over the space of metrics modulo diffeomorphisms we show that the anomaly bundle for a pair of self dual fields differs from the determinant bundle of the dirac operator coupled to chiral spinors by a flat bundle that is not trivial if the underlying manifold has middle degree cohomology and whose holonomies are determined explicitly we briefly sketch the relevance of this result for the computation of the global gravitational anomaly of the self dual field theory that will appear in another paper

the second homology group of the mapping class group of an orientable surface in i 7 mumford shows that the picard group p ic is isomorphic to h2 f 2 and conjectures the latter is rank one g 3 we prove this below for g 5 another interpretation of this theorem may be obtained by identifying h2 f as bordism classes of fiber bundles f w4 t where t is a closed oriented surface sect 0 when f is closed every such bundle is bordant to f w t a bundle admitting a section s t w the theorem then says that

the birational type of the moduli space of even spin curves abstract we determine the kodaira dimension of the moduli space s g of even spin curves for all g precisely we show that s g is of general type for g 8 and has negative kodaira dimension for g 8

deformations of 3 algebras we phrase deformations of n leibniz algebras in terms of the cohomology theory of the associated leibniz algebra we do the same for n lie algebras and for the metric versions of n leibniz and n lie algebras we place particular emphasis on the case of n 3 and explore the deformations of 3 algebras of relevance to three dimensional superconformal chern simons theories with matter

picard vessiot theory and ziglin s theorem abstract given a hamiltonian system ziglin s theorem is one of the criteria used to study the integrability of the system for this we make use of the variational equations along a known solution in this paper we obtain some consequences of the application of picard vessiot differential galois theory to ziglin s theorem first we present the general results for the case of a finite arbitrary number of degrees of freedom then we pass to the case of two degrees of freedom where the results can be given with more detail under the hypothesis of ziglin s theorem and some additional technical assumptions the main results theorems 1 and 2 relate the integrability of the hamiltonian with the properties of the differential galois group of the picard vessiot extension associated to the normal reduced variational equations for two degrees of freedom it is possible to study also the resonant case theorems 4 and 5 give the full classification of the galois groups and the related extensions in the integrable case a couple of applications are made to recover ito s theorem and to study the lame equation

non integrability of the fourth order truncated toda hamiltonian abstract the integrability of a polynomial hamiltonian derived through the fourth order truncation of the three particle toda lattice is investigated a theorem by ziglin is used to prove that this hamiltonian does not possess a second analytic integral although earlier numerical calculations had suggested its integrability some generalization of this system is also discussed

the level four braid group by evaluating the burau representation at t 1 we obtain a symplectic representation of the braid group we study the resulting congruence subgroups of the braid group namely the preimages of the principal congruence subgroups of the symplectic group our main result is that the level 4 congruence subgroup is equal to the group generated by squares of dehn twists we also show that the image of the brunnian subgroup of the braid group under the symplectic representation is the level four congruence subgroup

the non integrability of the truncated two fixed centres problem in a recent paper published in journal of differential equations irigoyen 2 proved that any truncation of order 3 of the symmetric two fixedcentres problem is not completely integrable through meromorphic integrals the purpose of this note is to point out that in fact neither is any truncationof the asymmetric problem this is deduced from a result published nearlysimultaneously by the authors 1 who established the non integrability ofany truncation of the zonal satellite problem the problem of the two fixed centres with masses equal to m

the picard group of the moduli space of r spin riemann surfaces abstract an r spin riemann surface is a riemann surface equipped with a choice of r th root of the co tangent bundle we give a careful construction of the moduli space orbifold of r spin riemann surfaces and explain how to establish a madsen weiss theorem for it this allows us to prove the mumford conjecture for these moduli spaces but more interestingly allows us to compute their algebraic picard groups for g 10 or g 9 in the 2 spin case we give a complete description of these picard groups in terms of explicitly constructed line bundles

three algebras triple systems and 3 graded lie superalgebras the three algebras used by bagger and lambert in n 6 theories of abjm type are in one to one correspondence with a certain type of lie superalgebras we show that the description of three algebras as generalized jordan triple systems naturally leads to this correspondence furthermore we show that simple three algebras correspond to simple lie superalgebras and vice versa this gives a classification of simple three algebras from the well known classification of simple lie superalgebras

non integrability of homogeneous two dimensional hamiltonians with velocity dependent potential the non integrability of classes of homogeneous two dimensional hamiltonian systems with a polynomial velocity dependent potential is shown on the basis of ziglin s theorem an analytic expression for the trace of the relevant monodromy matrices is presented which fits the numerical data perfectly an application is made to fokker planck hamiltonians with quadratic and with cubic drift terms

the picard group of the moduli space of curves with level structures for 4 l and g large we calculate the integral picard groups of the moduli spaces of curves and principally polarized abelian varieties with level l structures in particular we determine the divisibility properties of the standard line bundles over these moduli spaces and we calculate the second integral cohomology group of the level l subgroup of the mapping class group in a previous paper the author determined this rationally this entails calculating the abelianization of the level l subgroup of the mapping class group generalizing previous results of perron sato and the author finally along the way we calculate the first homology group of the mod l symplectic group with coefficients in the adjoint representation

the second rational homology group of the moduli space of curves with level structures abstract let be a finite index subgroup of the mapping class group of a closed genus g surface that contains the torelli group for instance can be the level l subgroup or the spin mapping class group we show that h 2 q q for g 5 a corollary of this is that the rational picard groups of the associated finite covers of the moduli space of curves are equal to q we also prove analogous results for surface with punctures and boundary components

relative equilibria and bifurcations in the generalized van der waals 4d oscillator abstract a uniparametric 4 dof family of perturbed hamiltonian oscillators in 1 1 1 1 resonance is studied as a generalization for several models for perturbed keplerian systems normalization by lie transforms only first order is considered here as well as geometric reduction related to the invariants associated to the symmetries is used based on the previous work of the authors a description is given for the lower dimensional relative equilibria in such normalized systems for which we introduce in this context the new concept of moment polytope in addition bifurcations of relative equilibria corresponding to 3d tori are studied in some particular cases where we focus on hamiltonian hopf bifurcations and bifurcations in the 3d van der waals and zeeman systems

simplicity in the faulkner construction we revisit the faulkner construction of metric 3 leibniz algebras admitting an embedding lie super algebra in the case of positive definite signature we relate the various notions of simplicity of the 3 algebra of the representation and of the embedding lie super algebra this reduces their classification to the extant classifications of simple lie super algebras

non integrability of the truncated toda lattice hamiltonian at any order two sufficient conditions for the non existence of an additional analytic integral are given for hamiltonian systems with non homogeneous polynomial potential of an arbitrary degree an application is made to the truncated three particle toda lattice which is proved to be non integrable at any order

stability of the homology of the mapping class groups of orientable surfaces the mapping class group of f fgs r is f rgs wo a where a is the topological group of orientation preserving diffeomorphisms of f which are the identity on df and fix the s punctures when r 0 and 2g s 3 an important feature of f is that it acts properly discontinuously on the teichmiller space us the quotient is the moduli space f of isometry classes of complete hyperbolic metrics of finite area on f since us is homeomorphic to euclidean space and f is virtually torsion free we have

orbital behaviour transition from the he non heiles to the three particle toda lattice hamiltonian abstract we examine the orbital behaviour in successive n order approximations 3 n 10 of the three particle toda lattice n 3 corresponding to the henon heiles galactic hamiltonian and in transition models between the 3rd and 4th orders of this expansion by using the poincare consequent method and by computing the e folding rate of the main unstable periodic orbits we locate and quantify the most important instabilities responsible for chaos in systems corresponding to even and odd order expansions

superpotentials for superconformal chern simons theories from representation theory these notes provide a detailed account of the universal structure of superpotentials defining a large class of superconformal chern simons theories with matter many of which appear as the low energy descriptions of multiple m2 brane configurations the amount of superconformal symmetry in the chern simons matter theory determines the minimum amount of global symmetry that the associated quartic superpotential must realize which in turn restricts the matter superfield representations our analysis clarifies the necessary representation theoretic data which guarantees a particular amount of superconformal symmetry thereby we shall recover all the examples of m2 brane effective field theories that have appeared in the recent literature the results are based on a refinement of the unitary representation theory of lie algebras to the case when the lie algebra admits an ad invariant inner product the types of representation singled out by the superconformal symmetry turn out to be intimately associated with triple systems admitting embedding lie super algebras and we obtain a number of new results about these triple systems which might be of independent interest in particular we prove that any metric 3 lie algebra embeds into a real metric 3 graded lie superalgebra in such a way that the 3 bracket is given by a nested lie bracket

darboux points and integrability of hamiltonian systems with homogeneous polynomial potential in this paper we study the integrability of natural hamiltonian systems with a homogeneous polynomial potential the strongest necessary conditions for their integrability in the liouville sense have been obtained by a study of the differential galois group of variational equations along straight line solutions these particular solutions can be viewed as points of a projective space of dimension smaller by one than the number of degrees of freedom we call them darboux points we analyze in detail the case of two degrees of freedom we show that except for a radial potential the number of darboux points is finite and it is not greater than the degree of the potential moreover we analyze cases when the number of darboux points is smaller than maximal for two degrees of freedom the above mentioned necessary condition for integrability can be expressed in terms of one nontrivial eigenvalue of the hessian of potential calculated at a darboux point we prove that for a given potential these nontrivial eigenvalues calculated for all darboux points cannot be arbitrary because they satisfy a certain relation which we give in an explicit form we use this fact to strengthen maximally the necessary conditions for integrability and we show that in a generic case for a given degree of the potential there is only a finite number of potentials which satisfy these conditions we also describe the nongeneric cases as an example we give a full list of potentials of degree four satisfying these conditions then investigating the differential galois group of higher order variational equations we prove that except for one discrete family among these potentials only those which are already known to be integrable are integrable we check that a finite number of potentials from the exceptional discrete family are not integrable and we conjecture that all of them are not integrable

on the non integrability of some generalized toda lattices abstract we consider a one parameter family of two degrees of freedom generalized toda hamiltonians using ziglin s theorem with some extensions we prove the non existence of an additional single valued analytic integral except for special values of the parameter

signs and approximate magnitudes of lyapunov exponents in continuous time dynamical systems methods for algebraically determining the signs and the magnitudes of lyapunov exponents of a given dynamical system are studied the existence of zero lyapunov exponents for the toda henon heiles and rossler systems are shown the approximate lyapunov spectra of lorenz and rossler systems are computed

on the non integrability of the stark zeeman hamiltonian system in this paper we present a proof of the non integrability in the liouvillian sense of the stark zeeman hamiltonian in particular we generalize the result of kummer and saenz about the non integrability of the pure zeeman hamiltonian the proof we give is an application of the theorem of morales and ramis 1998 about non integrability based on differential galois theory

every mapping class group is generated by 6 involutions abstract let mod g b denote the mapping class group of a surface of genus g with b punctures luo asked in torsion elements in the mapping class group of a surface math gt 0004048 v1 8apr2000 if there is a universal upper bound independent of genus for the number of torsion elements needed to generate mod g b we answer luo s question by proving that 3 torsion elements suffice to generate mod g 0 we also prove the more delicate result that there is an upper bound independent of genus not only for the number of torsion elements needed to generate mod g b but also for the order of those elements in particular our main result is that 6 involutions i e orientation preserving diffeomorphisms of order two suffice to generate mod g b for every genus g 3 b 0 and g 4 b 1

the structure of the torelli group i a finite set of generators for j this is the first of three papers concerning the so called torelli group let m mg be a compact orientable surface of genus g having n boundary components and let 9 xg be its mapping class group that is the group of orientation preserving diffeomorphisms of m which are 1 on the boundary am modulo isotopies which fix 3m pointwise this group is also known to the complex analysts as the teichmuller group or modular group if n 0 or 1 let further 4 jg be the subgroup of d1 which acts trivially on h1 m z the topologists have no traditional name for a but the analysts tell me it was known classically and is called the torelli group several interesting problems and conjectures exist concerning f the principal one can be found in kirby s problem list k and asks if gg is finitely generated in this first paper we shall answer the question affirmatively for both gg 0 and fgg when g 3 and shall give a fairly simple set of generators two other conjectures were made by the author the first involves the subgroup j of f which is generated by twists on nulhomologous simple closed curves ji produces a surjective homomorphism t fgg 1 a3h1 m z which kills c and it is conjectured there that in fact 5y ker t the proof of this is the content of the second paper in the third paper we use the results of the second to compute the abelianization f f explicitly thereby verifying another conjecture in ji the first reasonably simple but infinite set of generators for fg0 was produced by powell in p his generators consist of two types a twists on bounding simple closed curves b opposite twists on a bounding pair of disjoint homologous simple closed curves each of which are nonbounding using his result the author showed in j2 that the maps of the second type which we call bp maps for bounding pair are actually sufficient to generate both kg 0 and fg 1 for g 3 and in fact that we need only those whose two curves bound a genus one subsurface of m note that for g 2 all bp maps are trivial and hence the result fails in this case in the finite set of generators produced in this paper only

on the nonintegrability of the generalized van der waals hamiltonian system in this paper we aim to prove that except for the three known cases the uniparametric family of hamiltonian systems defined by the generalized van der waals potential is nonintegrable in the liouville arnold sense the proof is based on the theorem of morales and ramis about nonintegrability by differential galois theory

nonintegrability of nonhomogeneous nonlinear lattices we study the integrability of nonlinear lattices with nonhomogeneous polynomial potentials we prove a nonintegrability theorem for these dynamical systems

the tensor hierarchy algebra we introduce an infinite dimensional lie superalgebra which is an extension of the u duality lie algebra of maximal supergravity in d dimensions for 3 d 7 the level decomposition with respect to the u duality lie algebra gives exactly the tensor hierarchy of representations that arises in gauge deformations of the theory described by an embedding tensor for all positive levels p we prove that these representations are always contained in those coming from the associated borcherds kac moody superalgebra and we explain why some of the latter representations are not included in the tensor hierarchy the most remarkable feature of our lie superalgebra is that it does not admit a triangular decomposition like a borcherds kac moody super algebra instead the hodge duality relations between level p and d 2 p extend to negative p relating the representations at the first two negative levels to the supersymmetry and closure constraints of the embedding tensor

equivariant lefschetz maps for simplicial complexes and smooth manifolds let x be a locally compact space with a continuous proper action of a locally compact group g assuming that x satisfies a certain kind of duality in equivariant bivariant kasparov theory we can enrich the classical construction of lefschetz numbers for self maps to an equivariant k homology class we compute the lefschetz invariants for self maps of finite dimensional simplicial complexes and smooth manifolds the resulting invariants are independent of the extra structure used to compute them since smooth manifolds can be triangulated we get two formulas for the same lefschetz invariant in this case the resulting identity is closely related to the equivariant lefschetz fixed point theorem of luck and rosenberg

a criterion for the non existence of an additional integral in hamiltonian systems with a homogeneous potential abstract a non integrability criterion on the basis of ziglin s theorem is given for two degrees of freedom hamiltonians with a homogeneous potential of integer degree as simple examples the anisotropic kepler problem and the one dimensional newtonian three body problem are proved to be non integrable a self contained proof of ziglin s theorem n dimensional homogeneous potential version is also given

euler characteristics and gysin sequences for group actions on boundaries let g be a locally compact group let x be a universal proper g space and let open image in new window be a g equivariant compactification of x that is h equivariantly contractible for each compact subgroup open image in new window let open image in new window assuming the baum connes conjecture for g with coefficients open image in new window and c x we construct an exact sequence that computes the map on k theory induced by the embedding open image in new window this exact sequence involves the equivariant euler characteristic of x which we study using an abstract notion of poincare duality in bivariant k theory as a consequence if g is torsion free and the euler characteristic open image in new window is non zero then the unit element of open image in new window is a torsion element of order open image in new window furthermore we get a new proof of a theorem of luck and rosenberg concerning the class of the de rham operator in equivariant k homology

an equivariant parametric oka principle for bundles of homogeneous spaces we prove a parametric oka principle for equivariant sections of a holomorphic fibre bundle e with a structure group bundle mathscr g on a reduced stein space x such that the fibre of e is a homogeneous space of the fibre of mathscr g with the complexification k mathbb c of a compact real lie group k acting on x mathscr g and e our main result is that the inclusion of the space of k mathbb c hbox equivariant holomorphic sections of e over x into the space of k hbox equivariant continuous sections is a weak homotopy equivalence the result has a wide scope we describe several diverse special cases we use the result to strengthen heinzner and kutzschebauch s classification of equivariant principal bundles and to strengthen an oka principle for equivariant isomorphisms proved by us in a previous paper

n 5 three algebras and 5 graded lie superalgebras we discuss a generalization of n 6 three algebras to n 5 three algebras in connection to anti lie triple systems and basic lie superalgebras of type ii we then show that the structure constants defined in anti lie triple systems agree with those of n 5 superconformal theories in three dimensions

analytical non integrability of the truncated two fixed centres problem in the symmetric case abstract the problem of two fixed centres is an integrable hamiltonian system if one truncates the taylor expansion of the potential of this problem in the symmetric case at any order 3 we prove that one obtains a system which does not admit any first integral meromorphic and functionally independent of the energy and the angular momentum the proof is mainly founded on the criterion of non integrability for homogeneous potentials derived by yoshida from ziglin s theorem then we use this result to prove that the vinti problem truncated at any order 3 is analytically non integrable le probleme des deux centres fixes est un systeme hamiltonien integrable si l on tronque a un ordre arbitraire 3 le developpement de taylor du potentiel de ce probleme dans le cas symetrique on montre que le systeme obtenu n admet aucune integrale premiere meromorphe et fonctionnellement independante de l energie et du moment angulaire la demonstration est principalement fondee sur le critere de non integrabilite pour les potentiels homogenes que yoshida a deduit du theoreme de ziglin nous utilisons ensuite ce resultat pour etablir que le probleme de vinti tronque a un ordre arbitraire 3 est non integrable analytiquement

abelian quotients of mapping class groups of highly connected manifolds we compute the abelianisations of the mapping class groups of the manifolds w g 2n g s n times s n for n ge 3 and g ge 5 the answer is a direct sum of two parts the first part arises from the action of the mapping class group on the middle homology and takes values in the abelianisation of the automorphism group of the middle homology the second part arises from bordism classes of mapping tori and takes values in the quotient of the stable homotopy groups of spheres by a certain subgroup which in many cases agrees with the image of the stable j homomorphism we relate its calculation to a purely homotopy theoretic problem

generalized kac moody algebras on etudie une classe d algebres de lie qui ont une forme bilineaire contravariante qui est presque definie positive ces algebres generalisent celles de kac moody et peuvent etre considerees comme des algebres de kac moody avec des racines simples imaginaires

representation theory and homological stability abstract we introduce the idea of representation stability and several variations for a sequence of representations v n of groups g n a central application of the new viewpoint we introduce here is the importation of representation theory into the study of homological stability this makes it possible to extend classical theorems of homological stability to a much broader variety of examples representation stability also provides a framework in which to find and to predict patterns from classical representation theory littlewood richardson and murnaghan rules stability of schur functors to cohomology of groups pure braid torelli and congruence groups to lie algebras and their homology to the equivariant cohomology of flag and schubert varieties to combinatorics the n 1 n 1 conjecture the majority of this paper is devoted to exposing this phenomenon through examples in doing this we obtain applications theorems and conjectures beyond the discovery of new phenomena the viewpoint of representation stability can be useful in solving problems outside the theory in addition to the applications given in this paper it is applied by church ellenberg farb in preparation 20 to counting problems in number theory and finite group theory representation stability is also used by church 2012 19 to give broad generalizations and new proofs of classical homological stability theorems for configuration spaces on oriented manifolds

k leibniz algebras from lower order ones from lie triple to lie l ple systems two types of higher order lie l ple systems are introduced in this paper they are defined by brackets with l 3 arguments satisfying certain conditions and generalize the well known lie triple systems one of the generalizations uses a construction that allows us to associate a 2n 3 leibniz algebra l with a metric n leibniz algebra l by using a 2 n 1 linear kasymov trace form for l some specific types of k leibniz algebras relevant in the construction are introduced as well both higher order lie l ple generalizations reduce to the standard lie triple systems for l 3

differential geometric invariants for time reversal symmetric bloch bundles the real case topological quantum systems subjected to an even resp odd time reversal symmetry can be classified by looking at the related real resp quaternionic bloch bundles if from one side the topological classification of these time reversal vector bundle theories has been completely described in de nittis and gomi j geom phys 86 303 338 2014 for the real case and in de nittis and gomi commun math phys 339 1 55 2015 for the quaternionic case from the other side it seems that a classification in terms of differential geometric invariants is still missing in the literature with this article and its companion g de nittis and k gomi unpublished we want to cover this gap more precisely we extend in an equivariant way the theory of connections on principal bundles and vector bundles endowed with a time reversal symmetry in the real case we generalize the chern weil theory and we show that the assignment of a real connection along with the related differential chern class and its holonomy suffices for the classification of real vector bundles in low dimensions

on the lie algebraic origin of metric 3 algebras since the pioneering work of bagger lambert and gustavsson there has been a proliferation of three dimensional superconformal chern simons theories whose main ingredient is a metric 3 algebra on the other hand many of these theories have been shown to allow for a reformulation in terms of standard gauge theory coupled to matter where the 3 algebra does not appear explicitly in this paper we reconcile these two sets of results by pointing out the lie algebraic origin of some metric 3 algebras including those which have already appeared in three dimensional superconformal chern simons theories more precisely we show that the real 3 algebras of cherkis samann which include the metric lie 3 algebras as a special case and the hermitian 3 algebras of bagger lambert can be constructed from pairs consisting of a metric real lie algebra and a faithful real or complex respectively unitary representation this construction generalises and we will see how to construct many kinds of metric 3 algebras from pairs consisting of a real metric lie algebra and a faithful real complex or quaternionic unitary representation in the real case these 3 algebras are precisely the cherkis samann algebras which are then completely characterised in terms of this data in the complex and quaternionic cases they constitute generalisations of the bagger lambert hermitian 3 algebras and anti lie triple systems respectively which underlie n 6 and n 5 superconformal chern simons theories respectively in the process we rederive the relation between certain types of complex 3 algebras and metric lie superalgebras