Showing papers in "Journal of Mathematical Physics in 2010"
TL;DR: In this article, a general integrable coupled nonlinear Schrodinger system is investigated, where the coefficients of the self-phase modulation, cross-phase and four-wave mixing terms are more general while still maintaining integrability.
Abstract: In this paper, a general integrable coupled nonlinear Schrodinger system is investigated. In this system, the coefficients of the self-phase modulation, cross-phase modulation, and four-wave mixing terms are more general while still maintaining integrability. The N-soliton solutions in this system are obtained by the Riemann–Hilbert method. The collision dynamics between two solitons is also analyzed. It is shown that this collision exhibits some new phenomena (such as soliton reflection) which have not been seen before in integrable systems. In addition, the recursion operator and conservation laws for this system are also derived.
309 citations
TL;DR: In this article, it was shown that for any perturbation V representable as a sum of short-range bounded-norm interactions, the perturbed Hamiltonian H=H0+ϵV has well-defined spectral bands originating from low-lying eigenvalues of H0.
Abstract: We study zero-temperature stability of topological phases of matter under weak time-independent perturbations. Our results apply to quantum spin Hamiltonians that can be written as a sum of geometrically local commuting projectors on a D-dimensional lattice with certain topological order conditions. Given such a Hamiltonian H0, we prove that there exists a constant threshold ϵ>0 such that for any perturbation V representable as a sum of short-range bounded-norm interactions, the perturbed Hamiltonian H=H0+ϵV has well-defined spectral bands originating from low-lying eigenvalues of H0. These bands are separated from the rest of the spectra and from each other by a constant gap. The band originating from the smallest eigenvalue of H0 has exponentially small width (as a function of the lattice size). Our proof exploits a discrete version of Hamiltonian flow equations, the theory of relatively bounded operators, and the Lieb–Robinson bound.
290 citations
TL;DR: In this paper, an exact unitary transformation that maps the Hamiltonian of a quantum system coupled linearly to a continuum of bosonic or fermionic modes to a Hamiltonian that describes a one-dimensional chain with only nearest-neighbor interactions is presented.
Abstract: By using the properties of orthogonal polynomials, we present an exact unitary transformation that maps the Hamiltonian of a quantum system coupled linearly to a continuum of bosonic or fermionic modes to a Hamiltonian that describes a one-dimensional chain with only nearest-neighbor interactions. This analytical transformation predicts a simple set of relations between the parameters of the chain and the recurrence coefficients of the orthogonal polynomials used in the transformation and allows the chain parameters to be computed using numerically stable algorithms that have been developed to compute recurrence coefficients. We then prove some general properties of this chain system for a wide range of spectral functions and give examples drawn from physical systems where exact analytic expressions for the chain properties can be obtained. Crucially, the short-range interactions of the effective chain system permit these open-quantum systems to be efficiently simulated by the density matrix renormalization group methods.
270 citations
TL;DR: In this article, the existence of d2 equiangular lines in complex dimensions was studied and the authors provided numerical solutions in all dimensions d≤67 and, moreover, a putatively complete list of Weyl-Heisenberg covariant solutions for d ≥ 50.
Abstract: We report on a new computer study of the existence of d2 equiangular lines in d complex dimensions. Such maximal complex projective codes are conjectured to exist in all finite dimensions and are the underlying mathematical objects defining symmetric informationally complete measurements in quantum theory. We provide numerical solutions in all dimensions d≤67 and, moreover, a putatively complete list of Weyl–Heisenberg covariant solutions for d≤50. A symmetry analysis of this list leads to new algebraic solutions in dimensions d=24, 35, and 48, which are given together with algebraic solutions for d=4,…,15, and 19.
246 citations
TL;DR: In this paper, a fractional theory of the calculus of variations for multiple integrals is introduced, which uses the recent notions of Riemann-Liouville fractional derivatives and integrals in the sense of Jumarie.
Abstract: We introduce a fractional theory of the calculus of variations for multiple integrals. Our approach uses the recent notions of Riemann–Liouville fractional derivatives and integrals in the sense of Jumarie. The main results provide fractional versions of the theorems of Green and Gauss, fractional Euler–Lagrange equations, and fractional natural boundary conditions. As an application we discuss the fractional equation of motion of a vibrating string.
145 citations
TL;DR: In this article, the authors present a strong asymptotic stochastic flocking estimate for the stochastically perturbed Cucker-Smale model with multiplicative white noises.
Abstract: We present a strong asymptotic stochastic flocking estimate for the stochastically perturbed Cucker–Smale model. We characterize a form of multiplicative white noises and present sufficient conditions on the control parameters to guarantee the almost sure exponential convergence toward constant equilibrium states. When the strength of multiplicative noises is sufficiently large, we show that the strong stochastic flocking occurs even for negative communication weights.
142 citations
TL;DR: In this article, the authors present an algorithm for computing line bundle valued cohomology classes over toric varieties, which is the basic starting point for computing massless modes in both heterotic and type IIB/F theory compactifications.
Abstract: We present an algorithm for computing line bundle valued cohomology classes over toric varieties. This is the basic starting point for computing massless modes in both heterotic and type IIB/F-theory compactifications, where the manifolds of interest are complete intersections of hypersurfaces in toric varieties supporting additional vector bundles.
126 citations
TL;DR: In this paper, an operational interpretation for the 4-tangle as a type of residual entanglement was proposed, and the authors were able to find the class of maximally entangled four-qubits states, characterized by four real parameters.
Abstract: We find an operational interpretation for the 4-tangle as a type of residual entanglement, somewhat similar to the interpretation of the 3-tangle. Using this remarkable interpretation, we are able to find the class of maximally entangled four-qubits states which is characterized by four real parameters. The states in the class are maximally entangled in the sense that their average bipartite entanglement with respect to all possible bipartite cuts is maximal. We show that while all the states in the class maximize the average tangle, there are only a few states in the class that maximize the average Tsillas or Renyi α-entropy of entanglement. Quite remarkably, we find that up to local unitaries, there exists two unique states, one maximizing the average α-Tsallis entropy of entanglement for all α ⩾ 2, while the other maximizing it for all 0 < α ⩽ 2 (including the von-Neumann case of α = 1). Furthermore, among the maximally entangled four qubits states, there are only three maximally entangled states that have the property that for two, out of the three bipartite cuts consisting of two-qubits verses two-qubits, the entanglement is 2 ebits and for the remaining bipartite cut the entanglement between the two groups of two qubits is 1 ebit. The unique three maximally entangled states are the three cluster states that are related by a swap operator. We also show that the cluster states are the only states (up to local unitaries) that maximize the average α-Renyi entropy of entanglement for all α ⩾ 2.
114 citations
TL;DR: In this article, the authors studied sequences of long-range dependent random variables whose distributions have asymptotic power-law decay, and which are called (q,α)-stable distributions.
Abstract: The α-stable distributions introduced by Levy play an important role in probabilistic theoretical studies and their various applications, e.g., in statistical physics, life sciences, and economics. In the present paper we study sequences of long-range dependent random variables whose distributions have asymptotic power-law decay, and which are called (q,α)-stable distributions. These sequences are generalizations of independent and identically distributed α-stable distributions and have not been previously studied. Long-range dependent (q,α)-stable distributions might arise in the description of anomalous processes in nonextensive statistical mechanics, cell biology, finance. The parameter q controls dependence. If q=1 then they are classical independent and identically distributed with α-stable Levy distributions. In the present paper we establish basic properties of (q,α)-stable distributions and generalize the result of Umarov et al. [Milan J. Math. 76, 307 (2008)], where the particular case α=2,q∊[1,3) was considered, to the whole range of stability and nonextensivity parameters α∊(0,2] and q∊[1,3), respectively. We also discuss possible further extensions of the results that we obtain and formulate some conjectures.
112 citations
TL;DR: In this article, the relation between the kernels of the twisting maps and the trace function has been investigated and examples of Hom-Nambu-Lie algebras obtained using this construction are provided.
Abstract: The need to consider n -ary algebraic structures, generalizing Lie and Poisson algebras, has become increasingly important in physics, and it should therefore be of interest to study the mathematical concepts related to n -ary algebras. The purpose of this paper is to investigate ternary multiplications (as deformations of n -Lie structures) constructed from the binary multiplication of a Hom–Lie algebra, a linear twisting map, and a trace function satisfying certain compatibility conditions. We show that the relation between the kernels of the twisting maps and the trace function plays an important role in this context and provide examples of Hom–Nambu–Lie algebras obtained using this construction.
106 citations
TL;DR: In this article, the authors studied the Hilbert space space of N-valent SU(2) intertwiners with fixed total spin, which can be identified, at the classical level, with a space of convex polyhedra with N faces and fixed total boundary area.
Abstract: In this work, we study the Hilbert space space of N-valent SU(2) intertwiners with fixed total spin, which can be identified, at the classical level, with a space of convex polyhedra with N faces and fixed total boundary area. We show that this Hilbert space provides, quite remarkably, an irreducible representation of the U(N) group. This gives us therefore a precise identification of U(N) as a group of area-preserving diffeomorphisms of polyhedral spheres. We use this result to get new closed formulas for the black hole entropy in loop quantum gravity.
TL;DR: In this paper, the authors introduce quantum versions of the χ2-divergence, provide a detailed analysis of their properties, and apply them in the investigation of mixing times of quantum Markov processes.
Abstract: We introduce quantum versions of the χ2-divergence, provide a detailed analysis of their properties, and apply them in the investigation of mixing times of quantum Markov processes. An approach similar to the one presented in the literature for classical Markov chains is taken to bound the trace-distance from the steady state of a quantum processes. A strict spectral bound to the convergence rate can be given for time-discrete as well as for time-continuous quantum Markov processes. Furthermore, the contractive behavior of the χ2-divergence under the action of a completely positive map is investigated and contrasted to the contraction of the trace norm. In this context we analyze different versions of quantum detailed balance and, finally, give a geometric conductance bound to the convergence rate for unital quantum Markov processes.
TL;DR: In this paper, the authors focus on the one-dimensional infinite square well and show that the purported ground state, which is based on a piecewise approach, is not a solution of the fractional Schrodinger equation for the general fractional parameter α.
Abstract: A number of papers over the past eight years have claimed to solve the fractional Schrodinger equation for systems ranging from the one-dimensional infinite square well to the Coulomb potential to one-dimensional scattering with a rectangular barrier. However, some of the claimed solutions ignore the fact that the fractional diffusion operator is inherently nonlocal, preventing the fractional Schrodinger equation from being solved in the usual piecewise fashion. We focus on the one-dimensional infinite square well and show that the purported ground state, which is based on a piecewise approach, is definitely not a solution of the fractional Schrodinger equation for the general fractional parameter α. On a more positive note, we present a solution to the fractional Schrodinger equation for the one-dimensional harmonic oscillator with α=1.
TL;DR: Odake and Sasaki as mentioned in this paper provided analytic proofs for the shape invariance of two families of infinitely many exactly solvable one-dimensional quantum potentials, obtained by deforming the well-known radial oscillator potential or the Darboux-Poschl-Teller potential by a degree l (l=1,2,…) eigenpolynomial.
Abstract: We provide analytic proofs for the shape invariance of the recently discovered [Odake and Sasaki, Phys. Lett. B 679, 414 (2009)] two families of infinitely many exactly solvable one-dimensional quantum mechanical potentials. These potentials are obtained by deforming the well-known radial oscillator potential or the Darboux–Poschl–Teller potential by a degree l (l=1,2,…) eigenpolynomial. The shape invariance conditions are attributed to new polynomial identities of degree 3l involving cubic products of the Laguerre or Jacobi polynomials. These identities are proved elementarily by combining simple identities.
TL;DR: In this article, the authors obtained the analytic bound state energy spectra and the corresponding two-component upper and lower spinors of the two Dirac particles, in closed form, by means of the Nikiforov-Uvarov method.
Abstract: We give the approximate analytic solutions of the Dirac equation for the Rosen–Morse potential including the spin-orbit centrifugal term. In the framework of the spin and pseudospin symmetry concept, we obtain the analytic bound state energy spectra and the corresponding two-component upper and lower spinors of the two Dirac particles, in closed form, by means of the Nikiforov–Uvarov method. The special cases of the s-wave κ=±1 (l=l=0) Rosen–Morse potential, the Eckart-type potential, the PT-symmetric Rosen–Morse potential, and the nonrelativistic limits are briefly studied.
TL;DR: In this article, it was shown that for massive scalars with MS2 < 0 and for massive transverse vectors with MV2≤−2(D−1)H2, where D is the dimension of space-time and H is the Hubble parameter.
Abstract: Just because the propagator of some field obeys a de Sitter invariant equation does not mean it possesses a de Sitter invariant solution. The classic example is the propagator of a massless, minimally coupled scalar. We show that the same thing happens for massive scalars with MS2<0 and for massive transverse vectors with MV2≤−2(D−1)H2, where D is the dimension of space-time and H is the Hubble parameter. Although all masses in these ranges give infrared divergent mode sums, using dimensional regularization (or any other analytic continuation technique) to define the mode sums leads to the incorrect conclusion that de Sitter invariant solutions exist except at discrete values of the masses.
TL;DR: In this paper, the connections between three classes of theories: A, quiver matrix models, d=2 conformal A(r) Toda field theories, and d=4 N=2 supersymmetric conformal quiver gauge theories are explored.
Abstract: We explore the connections between three classes of theories: A, quiver matrix models, d=2 conformal A(r) Toda field theories, and d=4 N=2 supersymmetric conformal A(r) quiver gauge theories. In particular, we analyze the quiver matrix models recently introduced by Dijkgraaf and Vafa (unpublished) and make detailed comparisons with the corresponding quantities in the Toda field theories and the N=2 quiver gauge theories. We also make a speculative proposal for how the matrix models should be modified in order for them to reproduce the instanton partition functions in quiver gauge theories in five dimensions. (C) 2010 American Institute of Physics. [doi:10.1063/1.3449328]
TL;DR: In this paper, it was shown that polynomials associated with solutions of certain conditionally exactly solvable potentials obtained via unbroken as well as broken supersymmetry belong to the category of exceptional orthogonal polynomial.
Abstract: It is shown that polynomials associated with solutions of certain conditionally exactly solvable potentials obtained via unbroken as well as broken supersymmetry belong to the category of exceptional orthogonal polynomials. Some properties of such polynomials, e.g., recurrence relation, ladder operators, differential equations, etc., have been obtained.
TL;DR: In this paper, the authors proposed a proper quantization rule, ∫xAxBk(x)dx−∫x0Ax0Bk0(x),dx=nπ, where k(x)=2M[E−V(x)]/ℏ, and n is the number of nodes of wave function ψ(x).
Abstract: We propose proper quantization rule, ∫xAxBk(x)dx−∫x0Ax0Bk0(x)dx=nπ, where k(x)=2M[E−V(x)]/ℏ. The xA and xB are two turning points determined by E=V(x), and n is the number of the nodes of wave function ψ(x). We carry out the exact solutions of solvable quantum systems by this rule and find that the energy spectra of solvable systems can be determined only from its ground state energy. The previous complicated and tedious integral calculations involved in exact quantization rule are greatly simplified. The beauty and simplicity of the rule come from its meaning—whenever the number of the nodes of ϕ(x) or the number of the nodes of the wave function ψ(x) increases by 1, the momentum integral ∫xAxBk(x)dx will increase by π. We apply this proper quantization rule to carry out solvable quantum systems such as the one-dimensional harmonic oscillator, the Morse potential and its generalization, the Hulthen potential, the Scarf II potential, the asymmetric trigonometric Rosen–Morse potential, the Poschl–Teller ty...
TL;DR: In this paper, a short and semishort positive energy, unitary representations of the Osp(2N|4) superconformal group in three dimensions are discussed, consistent with character formulas for the SO(3,2) conformal group.
Abstract: Possible short and semishort positive energy, unitary representations of the Osp(2N|4) superconformal group in three dimensions are discussed. Corresponding character formulas are obtained, consistent with character formulas for the SO(3,2) conformal group, revealing long multiplet decomposition at unitarity bounds in a simple way. Limits, corresponding to reduction to various Osp(2N|4) subalgebras, are taken in the characters that isolate contributions from fewer states, at a given unitarity threshold, leading to considerably simpler formula. Via these limits, applied to partition functions, closed formula for the generating functions for numbers of BPS operators in the free field limit of superconformal U(n)×U(n) N=6 Chern–Simons theory and its supergravity dual are obtained in the large n limit. Partial counting of semishort operators is performed and consistency between operator counting for the free field and supergravity limits with long multiplet decomposition rules is explicitly demonstrated. Part...
TL;DR: Galilean conformal algebra (GCA) is a boundary realization of the Newton-Hooke string algebra in the bulk AdS as mentioned in this paper, where the string lies along the direction transverse to the boundary, and the worldsheet is AdS2.
Abstract: Galilean conformal algebra (GCA) is an Inonu–Wigner (IW) contraction of a conformal algebra, while Newton–Hooke string algebra is an IW contraction of an Anti-de Sitter (AdS) algebra, which is the isometry of an AdS space. It is shown that the GCA is a boundary realization of the Newton–Hooke string algebra in the bulk AdS. The string lies along the direction transverse to the boundary, and the worldsheet is AdS2. The one-dimensional conformal symmetry so(2,1) and rotational symmetry so(d) contained in the GCA are realized as the symmetry on the AdS2 string worldsheet and rotational symmetry in the space transverse to the AdS2 in AdSd+2, respectively. It follows from this correspondence that 32 supersymmetric GCAs can be derived as IW contractions of superconformal algebras, psu(2,2∣4), osp(8∣4), and osp(8∗∣4). We also derive less supersymmetric GCAs from su(2,2∣2), osp(4∣4), osp(2∣4), and osp(8∗∣2).
TL;DR: In this article, Kostenko et al. showed that the Hamiltonians HX,α,q with δ-type point interactions at the centers xk on the positive half line in terms of energy forms are self-adjoint if they are lower semibounded.
Abstract: We study the Hamiltonians HX,α,q with δ-type point interactions at the centers xk on the positive half line in terms of energy forms. We establish analogs of some classical results on operators Hq=−d2/dx2+q with locally integrable potentials q∊Lloc1(R+). In particular, we prove that the Hamiltonian HX,α,q is self-adjoint if it is lower semibounded. This result completes the previous results of Brasche [“Perturbation of Schrodinger Hamiltonians by measures—selfadjointness and semiboundedness,” J. Math. Phys. 26, 621 (1985)] on lower semiboundedness. Also we prove the analog of Molchanov’s discreteness criteria, Birman’s result on stability of a continuous spectrum, and investigate discreteness of a negative spectrum. In the recent paper [Kostenko, A. and Malamud, M., “1–D Schrodinger operators with local point interactions on a discrete set,” J. Differ. Equations 249, 253 (2010)], it was shown that the spectral properties of HX,α≔HX,α,0 correlate with the corresponding spectral properties of a certain clas...
TL;DR: In this paper, the same authors revisited the same model and repeated and extended the same construction paying particular attention to all the subtle mathematical points, including the crucial role of Riesz bases.
Abstract: In a recent paper, Trifonov suggested a possible explicit model of a PT-symmetric system based on a modification of the canonical commutation relation. Although being rather intriguing, in his treatment many mathematical aspects of the model have just been neglected, making most of the results of that paper purely formal. For this reason we are reconsidering the same model and we repeat and extend the same construction paying particular attention to all the subtle mathematical points. From our analysis the crucial role of Riesz bases clearly emerges. We also consider coherent states associated with the model.
TL;DR: In this paper, the relation between the notions of non-disturbance, joint measurability, and commutativity is analyzed and quantified by means of a semidefinite program.
Abstract: We consider pairs of discrete quantum observables (POVMs) and analyze the relation between the notions of nondisturbance, joint measurability, and commutativity. We specify conditions under which these properties coincide or differ—depending, for instance, on the interplay between the number of outcomes and the Hilbert space dimension or on algebraic properties of the effect operators. We also show that (non-)disturbance is, in general, not a symmetric relation and that it can be decided and quantified by means of a semidefinite program.
TL;DR: In this paper, a Riemannian material manifold is associated with the body, with a metric that explicitly depends on the temperature distribution, and a change in temperature corresponds to a change of the material metric.
Abstract: In this paper we formulate a geometric theory of thermal stresses. Given a temperature distribution, we associate a Riemannian material manifold to the body, with a metric that explicitly depends on the temperature distribution. A change in temperature corresponds to a change in the material metric. In this sense, a temperature change is a concrete example of the so-called referential evolutions. We also make a concrete connection between our geometric point of view and the multiplicative decomposition of deformation gradient into thermal and elastic parts. We study the stress-free temperature distributions of the finite-deformation theory using curvature tensor of the material manifold. We find the zero-stress temperature distributions in nonlinear elasticity. Given an equilibrium configuration, we show that a change in the material manifold, i.e., a change in the material metric will change the equilibrium configuration. In the case of a temperature change, this means that given an equilibrium configura...
TL;DR: The general formula for the optimal rate at which singlets can be distilled from any given noisy and arbitrarily correlated entanglement resource by means of local operations and classical communication (LOCC) is obtained by employing the quantum information spectrum method.
Abstract: We obtain the general formula for the optimal rate at which singlets can be distilled from any given noisy and arbitrarily correlated entanglement resource by means of local operations and classical communication (LOCC). Our formula, obtained by employing the quantum information spectrum method, reduces to that derived by Devetak and Winter [Proc. R. Soc. London, Ser. A 461, 207 (2005)], in the special case of an independent and identically distributed resource. The proofs rely on a one-shot version of the so-called “hashing bound,” which, in turn, provides bounds on the one-shot distillable entanglement under general LOCC.
TL;DR: In this article, an extension of the usual axioms of quantization in which classical Nambu-Poisson structures are translated to n-Lie algebras at quantum level is described.
Abstract: We investigate the geometric interpretation of quantized Nambu–Poisson structures in terms of noncommutative geometries. We describe an extension of the usual axioms of quantization in which classical Nambu–Poisson structures are translated to n-Lie algebras at quantum level. We demonstrate that this generalized procedure matches an extension of Berezin–Toeplitz quantization yielding quantized spheres, hyperboloids, and superspheres. The extended Berezin quantization of spheres is closely related to a deformation quantization of n-Lie algebras as well as the approach based on harmonic analysis. We find an interpretation of Nambu–Heisenberg n-Lie algebras in terms of foliations of Rn by fuzzy spheres, fuzzy hyperboloids, and noncommutative hyperplanes. Some applications to the quantum geometry of branes in M-theory are also briefly discussed.
TL;DR: In this article, the authors studied the near field and scattering resonances of a double array of subwavelength dielectric cylinders in TM polarization and proved the existence of bound states at specific distances between the arrays in the spectral region.
Abstract: Electromagnetic bound states in the radiation continuum are studied for periodic double arrays of subwavelength dielectric cylinders in TM polarization. They are similar to localized waveguide mode solutions of Maxwell’s equations for metal cavities or defects of photonic crystals, but, in contrast to the latter, their spectrum lies in the radiation continuum. The phenomenon is identical to the existence of bound states in the radiation continuum in quantum mechanics, discovered by von Neumann and Wigner. In the formal scattering theory, these states appear as resonances with the vanishing width. For the system studied, the bound states are shown to exist at specific distances between the arrays in the spectral region where one or two diffraction channels are open. Analytic solutions are obtained for all bound states (below the radiation continuum and in it) in the limit of thin cylinders (the cylinder radius is much smaller than the wavelength). The existence of bound states is also established in the spectral region where three and more diffraction channels are open, provided the dielectric constant and radius of the cylinders are fine-tuned. The near field and scattering resonances of the structure are investigated when the distance between the arrays varies in a neighborhood of its critical values at which the bound states are formed. In particular, it is shown that the near field in the scattering process becomes significantly amplified in specific regions of the array as the distance approaches its critical values. The effect may be used to control optical nonlinear effects by varying the distance between the arrays near its critical values.
TL;DR: In this article, isolated and embedded eigenvalues in the generalized eigenvalue problem defined by two self-adjoint operators with a positive essential spectrum and a finite number of isolated eigen values were studied.
Abstract: We study isolated and embedded eigenvalues in the generalized eigenvalue problem defined by two self-adjoint operators with a positive essential spectrum and a finite number of isolated eigenvalues. The generalized eigenvalue problem determines the spectral stability of nonlinear waves in infinite-dimensional Hamiltonian systems. The theory is based on Pontryagin’s invariant subspace theorem and extends beyond the scope of earlier papers of Pontryagin, Krein, Grillakis, and others. Our main results are (i) the number of unstable and potentially unstable eigenvalues equals the number of negative eigenvalues of the self-adjoint operators, (ii) the total number of isolated eigenvalues of the generalized eigenvalue problem is bounded from above by the total number of isolated eigenvalues of the self-adjoint operators, and (iii) the quadratic forms defined by the two self-adjoint operators are strictly positive on the subspace related to the continuous spectrum of the generalized eigenvalue problem. Applicatio...
TL;DR: In this article, a construction of 3-Lie algebras with trace forms is presented, and the structure of the 3-lie algebras constructed from general linear Lie algesbras is studied.
Abstract: 3-Lie algebras have close relationships with many important fields in mathematics and mathematical physics. In this paper, we provide a construction of 3-Lie algebras in terms of Lie algebras and certain linear functions. Moreover, with the construction from γ-matrices and two-dimensional extensions of metric Lie algebras, all the complex 3-Lie algebras in dimension ≤5 are obtained along this approach. As a special case, we study the structure of the 3-Lie algebras constructed from the general linear Lie algebras with trace forms and prove that they are semisimple and local.