Jacobi Identity Research Articles

A general theory for gauge-free lifting

A theory for lifting equations of motion for charged particle dynamics, subject to given electromagnetic like forces, up to a gauge-free system of coupled Hamiltonian Vlasov-Maxwell like equations is given. The theory provides very general expressions for the polarization and magnetization vector fields in terms of the particle dynamics description of matter. Thus, as is common in plasma physics, the particle dynamics replaces conventional constitutive relations for matter. Several examples are considered including the usual Vlasov-Maxwell theory, a guiding center kinetic theory, Vlasov-Maxwell theory with the inclusion of spin, and a Vlasov-Maxwell theory with the inclusion of Dirac's magnetic monopoles. All are shown to be Hamiltonian field theories and the Jacobi identity is proven directly.

Even spin minimal model holography

The even spin W^e_\infty algebra that is generated by the stress energy tensor together with one Virasoro primary field for every even spin s \geq 4 is analysed systematically by studying the constraints coming from the Jacobi identities. It is found that the algebra is characterised, in addition to the central charge, by one free parameter that can be identified with the self-coupling constant of the spin 4 field. We show that W^e_\infty can be thought of as the quantisation of the asymptotic symmetry algebra of the even higher spin theory on AdS_3. On the other hand, W^e_\infty is also quantum equivalent to the so(N) coset algebras, and thus our result establishes an important aspect of the even spin minimal model holography conjecture. The quantum equivalence holds actually at finite central charge, and hence opens the way towards understanding the duality beyond the leading 't Hooft limit.

On the compact real forms of the lie algebras of type E 6 and F 4

We give a construction of the compact real form of the Lie algebra of type E6, using the finite irreducible subgroup of shape 33+3: SL3(3), which is isomorphic to a maximal subgroup of the orthogonal group Ω7(3). In particular we show that the algebra is uniquely determined by this subgroup. Conversely, we prove from first principles that the algebra satisfies the Jacobi identity, and thus give an elementary proof of existence of a Lie algebra of type E6. The compact real form of F4 is exhibited as a subalgebra.

2-Plectic geometry, Courant algebroids, and categorified prequantization

A 2-plectic manifold is a manifold equipped with a closed nondegenerate 3-form, just as a symplectic manifold is equipped with a closed nondegenerate 2-form. In 2-plectic geometry one finds the higher analogues of many structures familiar from symplectic geometry. For example, any 2-plectic manifold has a Lie 2-algebra consisting of smooth functions and Hamiltonian 1-forms. This is equipped with a Poisson-like bracket which only satisfies the Jacobi identity up to “coherent chain homotopy”. Over any 2-plectic manifold is a vector bundle equipped with extra structure called an exact Courant algebroid. This Courant algebroid is the 2-plectic analogue of a transitive Lie algebroid over a symplectic manifold. Its space of global sections also forms a Lie 2-algebra. We show that this Lie 2-algebra contains an important sub-Lie 2-algebra which is isomorphic to the Lie 2-algebra of Hamiltonian 1-forms. Furthermore, we prove that it is quasi-isomorphic to a central extension of the (trivial) Lie 2-algebra of Hamiltonian vector fields, and therefore is the higher analogue of the well-known Kostant–Souriau central extension in symplectic geometry. We interpret all of these results within the context of a categorified prequantization procedure for 2-plectic manifolds. In doing so, we describe how $U(1)$-gerbes, equipped with a connection and curving, and Courant algebroids are the 2-plectic analogues of principal $U(1)$ bundles equipped with a connection and their associated Atiyah Lie algebroids.

The [formula omitted] Lie superbialgebras

By direct calculations of matrix form of super Jacobi and mixed super Jacobi identities which are obtained from adjoint representation, and using the automorphism supergroup of the gl(1|1) Lie superalgebra, we determine and classify all gl(1|1) Lie superbialgebras. Then, by calculating their classical r-matrices, the gl(1|1) coboundary Lie superbialgebras and their types (triangular, quasi-triangular or factorizable) are determined, furthermore in this way super Poisson structures on the GL(1|1) Lie supergroup are obtained. Also, we classify Drinfeld superdoubles based on the gl(1|1) as a theorem. Afterwards, as a physical application of the coboundary Lie superbialgebras, we construct a new integrable system on the homogeneous superspace OSp(1|2)/U(1). Finally, we make use of the Lyakhovsky and Mudrov formalism in order to build up the deformed gl(1|1) Lie superalgebra related to all gl(1|1) coboundary Lie superbialgebras. For one case, the quantization at the supergroup level is also provided, including its quantum R-matrix.

Constraints and spectra of a deformed quantum mechanics

We examine a deformed quantum mechanics in which the commutator between coordinates and momenta is a function of momenta. The Jacobi identity constraint on a two-parameter class of such modified commutation relations (MCR's) shows that they encode an intrinsic maximum momentum; a sub-class of which also imply a minimum position uncertainty. Maximum momentum causes the bound state spectrum of the one-dimensional harmonic oscillator to terminate at finite energy, whereby classical characteristics are observed for the studied cases. We then use a semi-classical analysis to discuss general concave potentials in one dimension and isotropic power-law potentials in higher dimensions. Among other conclusions, we find that in a subset of the studied MCR's, the leading order energy shifts of bound states are of opposite sign compared to those obtained using string-theory motivated MCR's, and thus these two cases are more easily distinguishable in potential experiments.

Generalized Jacobi Identities and Ball-Box Theorem for Horizontally Regular Vector Fields

We consider a family of vector fields and we assume a horizontal regularity on their derivatives. We discuss the notion of commutator showing that different definitions agree. We apply our results to the proof of a ball-box theorem and Poincar\'e inequality for nonsmooth H\"ormander vector fields.

A doubly-refined enumeration of alternating sign matrices and descending plane partitions

It was shown recently by the authors that, for any n, there is equality between the distributions of certain triplets of statistics on n×n alternating sign matrices (ASMs) and descending plane partitions (DPPs) with each part at most n. The statistics for an ASM A are the number of generalized inversions in A, the number of −1ʼs in A and the number of 0ʼs to the left of the 1 in the first row of A, and the respective statistics for a DPP D are the number of nonspecial parts in D, the number of special parts in D and the number of nʼs in D. Here, the result is generalized to include a fourth statistic for each type of object, where this is the number of 0ʼs to the right of the 1 in the last row of an ASM, and the number of (n−1)ʼs plus the number of rows of length n−1 in a DPP. This generalization is proved using the known equality of the three-statistic generating functions, together with relations which express each four-statistic generating function in terms of its three-statistic counterpart. These relations are obtained by applying the Desnanot–Jacobi identity to determinantal expressions for the generating functions, where the determinants arise from standard methods involving the six-vertex model with domain-wall boundary conditions for ASMs, and nonintersecting lattice paths for DPPs.

Hamiltonian structure of reduced fluid models for plasmas obtained from a kinetic description

We consider the Hamiltonian structure of reduced fluid models obtained from a kinetic description of collisionless plasmas by Vlasov–Maxwell equations. We investigate the possibility of finding Poisson subalgebras associated with fluid models starting from the Vlasov–Maxwell Poisson algebra. In this way, we show that the only possible Poisson subalgebra involves the moments of zeroth and first order of the Vlasov distribution, meaning the fluid density and the fluid velocity. We find that the bracket derived in [B.A. Shadwick, G.M. Tarkenton, E.H. Esarey, Phys. Rev. Lett. 93 (2004) 175002] which involves moments of order 2 is not a Poisson bracket since it does not satisfy the Jacobi identity.

Bianchi identities for non‐geometric fluxes from quasi‐Poisson structures to Courant algebroids

AbstractStarting from a (non‐associative) quasi‐Poisson structure, the derivation of a Roytenberg‐type algebra is presented. From the Jacobi identities of the latter, the most general form of Bianchi identities for fluxes (H, f, Q, R) is then derived. It is also explained how this approach is related to the mathematical theory of quasi‐Lie and Courant algebroids.

Palatini–Lovelock–Cartan gravity—Bianchi identities for stringy fluxes

A Palatini-type action for Einstein and Gauss–Bonnet gravity with non-trivial torsion is proposed. A three-form flux is incorporated via a deformation of the Riemann tensor, and consistency of the Palatini variational principle requires the flux to be covariantly constant and to satisfy a Jacobi identity. Studying gravity actions of third order in the curvature leads to a conjecture about general Palatini–Lovelock–Cartan gravity. We point out potential relations to string-theoretic Bianchi identities and, using the Schouten–Nijenhuis bracket, derive a set of Bianchi identities for the non-geometric Q- and R-fluxes which include derivative and curvature terms. Finally, the problem of relating torsional gravity to higher order corrections of the bosonic string-effective action is revisited.

Automorphisms of real four dimensional Lie algebras

We are calculating the automorphisms of real Lie algebras in four dimensional matrix method. At first, we introduced the adjoint matrices and we reduced Jacobi and mix Jacobs identities to matrix form, then we presented the automorphisms of real lie algebras. At the end we compared our results to another article. Key words: Lie algebras, Lie brackets, automorphism, adjoint representations.

Combined Wronskian solutions to the 2D Toda molecule equation

By combining two pieces of bi-directional Wronskian solutions, molecule solutions in Wronskian form are presented for the finite, semi-infinite and infinite bilinear 2D Toda molecule equations. In the cases of finite and semi-infinite lattices, separated-variable boundary conditions are imposed. The Jacobi identities for determinants are the key tool employed in the solution formulations.

Simple Deformed Witt Algebras

For any unique factorization domain [Formula: see text] and an algebra endomorphism σ of [Formula: see text], there exists a non-associative algebra [Formula: see text] with multiplication satisfying skew-symmetry and generalized (twisted) Jacobi identities, called a σ-deformed Witt algebra. In this paper, we obtain necessary and sufficient conditions for the algebra [Formula: see text] to be simple.

Extensions of the Poincaré group

We construct an extension of the Poincare group which involves a mixture of internal and space-time supersymmetries. The resulting group is an extension of the superPoincare group with infinitely many generators which carry internal and space-time indices. It is a closed algebra since all Jacobi identities are satisfied and it has therefore explicit matrix representations. We investigate the massless case and construct the irreducible representations of the extended symmetry. They are divided into two sets, longitudinal and transversal representations. The transversal representations involve an infinite series of integer and half-integer helicities. Finally we suggest an extension of the conformal group along the same line.

Explicit BCJ numerators from pure spinors

We derive local kinematic numerators for gauge theory tree amplitudes which manifestly satisfy Jacobi identities analogous to color factors. They naturally emerge from the low energy limit of superstring amplitudes computed with the pure spinor formalism. The manifestation of the color-kinematics duality is a consequence of the superstring computation involving no more than (n − 2)! kinematic factors for the full color dressed n point amplitude. The bosonic part of these results describe gluon scattering independent on the number of supersymmetries and captures any NkMHV helicity configuration after dimensional reduction to D = 4 dimensions.

Wronskian and Grammian solutions to a (3 + 1)-dimensional generalized KP equation

Abstract Wronskian and Grammian formulations are established for a (3 + 1)-dimensional generalized KP equation, based on the Plucker relation and the Jacobi identity for determinants. Generating functions for matrix entries satisfy a linear system of partial differential equations involving a free parameter. Examples of Wronskian and Grammian solutions are computed and a few particular solutions are plotted.

L ∞-Algebras from Multisymplectic Geometry

A manifold is multisymplectic, or more specifically n-plectic, if it is equipped with a closed nondegenerate differential form of degree n+1. In our previous work with Baez and Hoffnung, we described how the `higher analogs' of the algebraic and geometric structures found in symplectic geometry should naturally arise in 2-plectic geometry. In particular, we showed that just as a symplectic manifold gives a Poisson algebra of functions, any 2-plectic manifold gives a Lie 2-algebra of 1-forms and functions. Lie n-algebras are examples of L-infinity algebras: graded vector spaces which are equipped with a collection of skew-symmetric multi-brackets that satisfy a generalized Jacobi identity. Here, we generalize our previous result. Given an n-plectic manifold, we explicitly construct a corresponding Lie n-algebra on a complex consisting of differential forms whose multi-brackets are specified by the n-plectic structure. We also show that any n-plectic manifold gives rise to another kind of algebraic structure known as a differential graded Leibniz algebra. We conclude by describing the similarities between these two structures within the context of an open problem in the theory of strongly homotopy algebras.

Classical mechanics on noncommutative space with Lie-algebraic structure

We investigate the kinetics of a nonrelativistic particle interacting with a constant external force on a Lie-algebraic noncommutative space. The structure constants of a Lie algebra, also called noncommutative parameters, are constrained in general due to some algebraic properties, such as the antisymmetry and Jacobi identity. Through solving the constraint equations the structure constants satisfy, we obtain two new sorts of algebraic structures, each of which corresponds to one type of noncommutative spaces. Based on such types of noncommutative spaces as the starting point, we analyze the classical motion of the particle interacting with a constant external force by means of the Hamiltonian formalism on a Poisson manifold. Our results not only include that of a recent work as our special cases, but also provide new trajectories of motion governed mainly by marvelous extra forces. The extra forces with the unimaginable t x ˙ - , ( xx ) ˙ -, and ( xx ) ¨ -dependence besides with the usual t-, x-, and x ˙ -dependence, originating from a variety of noncommutativity between different spatial coordinates and between spatial coordinates and momenta as well, deform greatly the particle’s ordinary trajectories we are quite familiar with on the Euclidean (commutative) space.

Projective geometry from Poisson algebras

By analogy with the Poisson algebra of quadratic forms on the symplectic plane and with the concept of duality in the projective plane introduced by Arnold (2005) [1], where the concurrence of the triangle altitudes is deduced from the Jacobi identity, we consider the Poisson algebras of the first degree harmonics on the sphere, on the pseudo-sphere and on the hyperboloid, to obtain analogous duality concepts and similar results for spherical, pseudo-spherical and hyperbolic geometry. Such algebras, including the algebra of quadratic forms, are isomorphic either to the Lie algebra of the vectors in R 3 , with the vector product, or to algebra s l 2 ( R ) . The Tomihisa identity, introduced in (Tomihisa, 2009) [3] for the algebra of quadratic forms, holds for all these Poisson algebras and has a geometrical interpretation. The relationships between the different definitions of duality in projective geometry inherited by these structures are shown here.