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Pre-Courant algebroids are `Courant algebroids' without the Jacobi identity for the Courant-Dorfman bracket. In this paper we examine the corresponding supermanifold description of pre-Courant algebroids and some direct consequences thereof - such as the definition of (sub-)Dirac structures and the notion of the naive quasi-cochain complex. In particular we define symplectic almost Lie 2-algebroids and show how they correspond to pre-Courant algebroids. Moreover, the framework of supermanifolds allows us to economically define and work with pre-Courant algebroids equipped with a compatible non-negative grading. VB-Courant algebroids are natural examples of what we call weighted pre-Courant algebroids, and our approach drastically simplifies working with them. We remark that examples of pre-Courant algebroids are plentiful - natural examples include the cotangent bundle of any almost Lie algebroid., Dedicated to the memory of James Alfred Bruce

We construct L∞ algebras for general “initial data” given by a vector space equipped with an antisymmetric bracket not necessarily satisfying the Jacobi identity. We prove that any such bracket can be extended to a 2-term L∞ algebra on a graded vector space of twice the dimension, with the 3-bracket being related to the Jacobiator. While these L∞ algebras always exist, they generally do not realize a nontrivial symmetry in a field theory. In order to define L∞ algebras with genuine field theory realizations, we prove the significantly more general theorem that if the Jacobiator takes values in the image of any linear map that defines an ideal there is a 3-term L∞ algebra with a generally nontrivial 4-bracket. We discuss special cases such as the commutator algebra of octonions, its contraction to the “R-flux algebra,” and the Courant algebroid.

Nambu dynamics is a generalized Hamiltonian dynamics of more than two variables, whose time evolutions are given by the Nambu bracket, a generalization of the canonical Poisson bracket. Nambu dynamics can always be represented in the form of noncanonical Hamiltonian dynamics by defining the noncanonical Poisson bracket by means of the Nambu bracket. For the time evolution to be consistent, the Nambu bracket must satisfy the fundamental identity, while the noncanonical Poisson bracket must satisfy the Jacobi identity. However, in many degrees of freedom systems, it is well known that the fundamental identity does not hold. In the present paper we show that, even if the fundamental identity is violated, the Jacobi identity for the corresponding noncanonical Hamiltonian dynamics could hold. As an example, we evaluate these identities for a semiclassical system of two coupled oscillators., 7 pages, final version

In this paper, the definition of the cross product of three tangent vectors to Open image in new window at the same point is stated, based on this definition a lemma is stated in which eight properties of said product are proposed and demonstrated. In addition, four theorems and two corollaries are stated and proved. One of the theorems constitutes an extension of the Jacobi identity. In some of the proofs, programs based on the paradigms: functional, rule-based and list-based from the Wolfram language, incorporated in Mathematica, are used.

We explore the double copy of effective field theories (EFTs), both in the (generalized) color-kinematics and Kawai-Lewellen-Tye (KLT) approaches. In the former, we define the linear map between color factors formed by single traces of generators and by products of the structure constants. It enables us to systematically construct scalar numerators satisfying the Jacobi identities from simpler numerator 'seeds' with trace-like permutation properties, with the advantage that this construction easily applies to any multiplicity. This map also relates the generalized KLT and color-kinematics formalisms, allowing to produce KLT kernels at arbitrary order in the EFT expansion. At 4-point, we show that all EFT kernels are generated and that they only yield double-copy amplitudes which can also be obtained from the traditional KLT kernel. We perform initial checks suggesting that the same conclusions also hold at 5-point. We focus on single-trace massless scalar EFTs which however also control the higher-derivative corrections to gauge and gravity theories.

The Hamiltonian structure of the guiding-center Vlasov-Maxwell equations is presented in terms of a Hamiltonian functional and a guiding-center Vlasov-Maxwell bracket. The bracket, which is shown to satisfy the Jacobi identity exactly, is used to show that the guiding-center momentum and angular-momentum conservation laws can also be expressed in Hamiltonian form., Comment: 6 pages. arXiv admin note: text overlap with arXiv:2107.08129

Classical Hamiltonian mechanics is realized by the action of a Poisson bracket on a Hamiltonian function. The Hamiltonian function is a constant of motion (the energy) of the system. The properties of the Poisson bracket are encapsulated in the symplectic 2-form, a closed second order differential form. Due to closure, the symplectic 2-form is preserved by the Hamiltonian flow, and it assigns an invariant (Liouville) measure on the phase space through the Lie-Darboux theorem. In this paper we propose a generalization of classical Hamiltonian mechanics to a three-dimensional phase space: the classical Poisson bracket is replaced with a generalized Poisson bracket acting on a pair of Hamiltonian functions, while the symplectic 2-form is replaced by a symplectic 3-form. We show that, using the closure of the symplectic 3-form, a result analogous to the classical Lie-Darboux theorem holds: locally, there exist smooth coordinates such that the components of the symplectic 3-form are constants, and the phase space is endowed with a preserved volume element. Furthermore, as in the classical theory, the Jacobi identity for the generalized Poisson bracket mathematically expresses the closure of the associated symplectic form. As a consequence, constant skew-symmetric third order contravariant tensors always define generalized Poisson brackets. This is in contrast with generalizations of Hamiltonian mechanics postulating the fundamental identity as replacement for the Jacobi identity. In particular, we find that the fundamental identity represents a stronger requirement than the closure of the symplectic 3-form., 16 pages

We study the problem of constructing a general hybrid quantum-classical bracket from a partial classical limit of a full quantum bracket. Introducing a hybrid composition product, we show that such a bracket is the commutator of that product. From this we see that the hybrid bracket will obey the Jacobi identity and the Leibniz rule provided the composition product is associative. This suggests that the set of hybrid variables belonging to an associative subalgebra with the composition product will have consistent quantum-classical dynamics. This restricts the class of allowed quantum-classical interaction Hamiltonians. Furthermore, we show that pure quantum or classical variables can interact in a consistent framework, unaffected by no-go theorems in the literature or the restrictions for hybrid variables. In the proposed scheme, quantum backreaction appears as quantum-dependent terms in the classical equations of motion., Published in Physical Review A, minor revisions to v3, 11 pages

In this study, we present an algorithm in MATLAB to classify $8$ dimensional non-abelian nilsoliton metric Lie algebras with singular Gram matrices. With this algorithm, we can compute Lie brackets, structure constants, index, rank and eigenvalues of the nilsoliton derivation. This paper can be considered as a follow up paper to our previous study \cite{Kad3}. In our previous paper, we defined an algorithm that helps to classify algebras that admit simple derivations and singular Gram matrices $U$. Since the Gram matrices are singular, there exists infinitely many solutions to $Uv=[1]_m$, where the solutions are exactly the structure constants’ squares. In order the algebra to be a Lie algebra, the structure constants has to satisfy the Jacobi identity. In our previous work, we did not present a methodology to classify algebras that satisfy Jacobi identity. But in this paper, we extend the capability in such a way that we are able to create and solve the Jacobi identity/identities with the help of computer algorithms for each index set. Therefore we completely classify all indecomposible nilsolitons in dimension $8$. Several examples are provided for the illustration of the methodology.

In this paper we provide a new graph theoretic proof of the tropical Jacobi identity, recently obtained in [AGN16]. We also develop an application of this theorem to optimal assignments with supervisions. That is, optimally assigning multiple tasks to one team, or daily tasks to multiple teams, where each team has a supervisor task or a supervised task., The first author was supported by INRIA postdoctoral fellowship and SCE Young research grant. The second author was supported by a public grant as part of the Investissement d'avenir project, reference ANR-11-LABX-0056-LMH, LabEx LHM. The third author was supported by EPSRC Grant EP/P019676/1

All gauge-natural bilinear operators $$A: \Gamma ^l_E(TE\oplus T^*E)\times \Gamma ^l_E(TE\oplus T^*E)\rightarrow \Gamma ^l_E(TE\oplus T^*E)$$A:ΓEl(TE⊕T∗E)×ΓEl(TE⊕T∗E)→ΓEl(TE⊕T∗E) transforming pairs of linear sections of the “doubled” tangent bundle $$TE\oplus T^*E$$TE⊕T∗E of a vector bundle E into linear sections of $$TE\oplus T^*E$$TE⊕T∗E are completely described. Then, all such A with the Jacobi identity in Leibniz form are extracted.

We compare three different ways of checking the Jacobi identity for weakly nonlocal Poisson brackets using the theory of distributions, of pseudodifferential operators and of Poisson vertex algebras, respectively. We show that the three approaches lead to similar computations and same results., 47 pages

We consider the quantization procedure and investigate the application of the quantizer–dequantizer method and star-product technique to construct associative products and the associative algebras formed by the quantizer–dequantizer operators and their symbols. The corresponding Lie algebras are also constructed. We study the case where the quantizer–dequantizer operators form a self-dual system and show that the structure constants of the Lie algebras satisfy some identity, in addition to the Jacobi identity. Using tomographic quantizer–dequantizer operators and their symbols, we construct the continuous associative algebra and the corresponding Lie algebra.

This paper is about the relationship of diagrams with mathematics. We study the linking of mathematical fields by examining first a diagrammatic for vector calculus, and then showing how it works and why it works by relating that formalism to the question of coloring maps and graphs in and out of the plane. The paper then considers knot diagrams and finally relationships with Lie algebras and the Jacobi identity.

We give a geometric interpretation of color-kinematics duality between tree-level scattering amplitudes of gauge and gravity theories. Using their representation as intersection numbers we show how to obtain Bern-Carrasco-Johansson numerators in a constructive way as residues around boundaries of the moduli space. In this language the kinematic Jacobi identity between each triple of numerators is a residue theorem in disguise., 5 pages, title change to match the published version

A triple vector bundle is a cube of vector bundle structures which commute in the (strict) categorical sense. A grid in a triple vector bundle is a collection of sections of each bundle structure with certain linearity properties. A grid provides two routes around each face of the triple vector bundle, and six routes from the base manifold to the total manifold; the warps measure the lack of commutativity of these routes. In this paper we first prove that the sum of the warps in a triple vector bundle is zero. The proof we give is intrinsic and, we believe, clearer than the proof using decompositions given earlier by one of us. We apply this result to the triple tangent bundle $$T^3M$$ of a manifold and deduce (as earlier) the Jacobi identity. We further apply the result to the triple vector bundle $$T^2A$$ for a vector bundle A using a connection in A to define a grid in $$T^2A$$ . In this case the curvature emerges from the warp theorem.

In this paper, we show certain series identities arising from the Jacobi identity of the ordinary theta function. These include several formulas of Ramanujan type given by Berndt and their relevant analogues.

We derive the quartic interaction vertex of pure Yang-Mills theory by demanding closure of the light-cone Poincar\'e algebra in four-dimensional Minkowski spacetime. This calculation explicitly shows why structure constants must satisfy the Jacobi identity. We then prove that corrections to the spin generator, for spin one at this order, vanish. We comment briefly on higher spin fields in this context., Comment: 10 pages. v2: minor changes, references added

We consider matrices over a commutative ring and characterize the class of outer inverses for which Jacobi type identities can be extended. We obtain a necessary and sufficient condition for the existence of Rao-regular Drazin inverse in terms of sum of principal minors of A k for some k. Also, we obtain determinantal formula for the Rao-regular Drazin inverse. Conjectures are formulated which give expressions for outer inverse and the conjectures are proved in some special cases.

We show that the replicator dynamics for zero-sum games arises as a result of a non-canonical bracket that is a hybrid between a Poisson Bracket and a Nambu Bracket. The resulting non-canonical bracket is parameterized both the by the skew-symmetric payoff matrix and a mediating function. The mediating function is only sometimes a conserved quantity, but plays a critical role in the determination of the dynamics. As a by-product, we show that for the replicator dynamics this function arises in the definition of a natural metric on which phase flow-volume is preserved. Additionally, we show that the non-canonical bracket satisfies all the same identities as the Poisson bracket except for the Jacobi identity (JI), which is satisfied for special cases of the mediating function. In particular, the mediating function that gives rise to the replicator dynamics yields a bracket that satisfies JI. This neatly explains why the mediating function allows us to derive a metric on which phase flow is conserved and suggests a natural geometry for zero-sum games that extends the Symplectic geometry of the Poisson bracket and potentially an alternate approach to quantizing evolutionary games., 13 pages, 2 figures