Jacobi Identity Research Articles

Operadic quantization as a tool for discrete geometry

The operadic Lax representations of the harmonic oscillator are used to construct the quantum counterparts of 3d real Lie algebras in the Bianchi classification. The Jacobi operators of these quantum algebras are studied. It is shown how the energy conservation is related to the Jacobi identity and how the quantization leads to an anomaly – the quantum violation of the Jacobi relations. By using the nonvanishing quantum Jacobi operators, the derivative quantum algebra for a triple of 3d real Lie algebras is defined. It is proposed that the derivative algebra is the 3d real Heisenberg algebra. From this it follows that in this model only the discrete values of the spatial coordinates are physically allowed.

Jacobi–Jordan algebras

We study finite-dimensional commutative algebras, which satisfy the Jacobi identity. Such algebras are Jordan algebras. We describe some of their properties and give a classification in dimensions n<7 over algebraically closed fields of characteristic not 2 or 3.

Phase space quantization, noncommutativity, and the gravitational field

In this paper we study the structure of the phase space in noncommutative geometry in the presence of a nontrivial frame. Our basic assumptions are that the underlying space is a symplectic and parallelizable manifold. Furthermore, we assume the validity of the Leibniz rule and the Jacobi identities. We consider noncommutative spaces due to the quantization of the symplectic structure and determine the momentum operators that guarantee a set of canonical commutation relations, appropriately extended to include the nontrivial frame. We stress the important role of left vs. right acting operators and of symplectic duality. This enables us to write down the form of the full phase space algebra on these noncommutative spaces, both in the noncompact and in the compact case. We test our results against the class of 4D and 6D symplectic nilmanifolds, thus presenting a large set of nontrivial examples that realize the general formalism.

Hamiltonian structure of a drift-kinetic model and Hamiltonian closures for its two-moment fluid reductions

We address the problem of the existence of the Hamiltonian structure for an electrostatic drift-kinetic model and for the related fluid models describing the evolution of the first two moments of the distribution function with respect to the parallel velocity. The drift-kinetic model, which accounts for background density and temperature gradients as well as polarization effects, is shown to possess a noncanonical Hamiltonian structure. The corresponding Poisson bracket is expressed in terms of the fluid moments and it is found that the set of functionals of the zero order moment forms a sub-algebra, thus automatically leading to a class of one-moment Hamiltonian fluid models. In particular, in the limit of weak spatial variations of the background quantities, the Charney-Hasegawa-Mima equation, with its Hamiltonian structure, is recovered. For the set of functionals of the first two moments, which, unlike the case of the Vlasov equation, turns out not to form a sub-algebra, we look for closures that lead to a closed Poisson bracket restricted to this set of functionals. The constraint of the Jacobi identity turns out to select the adiabatic equation of state for an ideal gas with one-degree-of-freedom molecules, as the only admissible closure in this sense. When the so called δf ordering is applied to the model, on the other hand, a Poisson bracket is obtained if the second order moment is a linear combination of the first two moments of the total distribution function. By means of this procedure, three-dimensional Hamiltonian fluid models that couple a generalized Charney-Hasegawa-Mima equation with an evolution equation for the parallel velocity are derived. Among these, a model adopted by Meiss and Horton [Phys. Fluids 26, 990 (1983)] to describe drift waves coupled to ion-acoustic waves, is obtained and its Hamiltonian structure is provided explicitly.

On realizations of polynomial algebras with three generators via deformed oscillator algebras

We present the most general polynomial Lie algebra generated by a second order integral of motion and one of order M, construct the Casimir operator, and show how the Jacobi identity provides the existence of a realization in terms of deformed oscillator algebra. We also present the classical analogue of this construction for the most general polynomial Poisson algebra. Two specific classes of such polynomial algebras are discussed that include the symmetry algebras observed for various 2D superintegrable systems.

Jacobi-type identities in algebras and superalgebras

We introduce two remarkable identities written in terms of single commutators and anticommutators for any three elements of an arbitrary associative algebra. One is a consequence of the other (fundamental identity). From the fundamental identity, we derive a set of four identities (one of which is the Jacobi identity) represented in terms of double commutators and anticommutators. We establish that two of the four identities are independent and show that if the fundamental identity holds for an algebra, then the multiplication operation in that algebra is associative. We find a generalization of the obtained results to the super case and give a generalization of the fundamental identity in the case of arbitrary elements. For nondegenerate even symplectic (super)manifolds, we discuss analogues of the fundamental identity.

ON THE UNIQUENESS OF HIGHER-SPIN SYMMETRIES IN ADS AND CFT

We study the uniqueness of higher-spin algebras which are at the core of higher-spin theories in AdS and of CFTs with exact higher-spin symmetry, i.e. conserved tensors of rank greater than two. The Jacobi identity for the gauge algebra is the simplest consistency test that appears at the quartic order for a gauge theory. Similarly, the algebra of charges in a CFT must also obey the Jacobi identity. These algebras are essentially the same. Solving the Jacobi identity under some simplifying assumptions listed out, we obtain that the Eastwood–Vasiliev algebra is the unique solution for d = 4 and d≥7. In 5d, there is a one-parameter family of algebras that was known before. In particular, we show that the introduction of a single higher-spin gauge field/current automatically requires the infinite tower of higher-spin gauge fields/currents. The result implies that from all the admissible non-Abelian cubic vertices in AdSd, that have been recently classified for totally symmetric higher-spin gauge fields, only one vertex can pass the Jacobi consistency test. This cubic vertex is associated with a gauge deformation that is the germ of the Eastwood–Vasiliev's higher-spin algebra.

Quantum mechanics with coordinate dependent noncommutativity

Noncommutative quantum mechanics can be considered as a first step in the construction of quantum field theory on noncommutative spaces of generic form, when the commutator between coordinates is a function of these coordinates. In this paper we discuss the mathematical framework of such a theory. The noncommutativity is treated as an external antisymmetric field satisfying the Jacobi identity. First, we propose a symplectic realization of a given Poisson manifold and construct the Darboux coordinates on the obtained symplectic manifold. Then we define the star product on a Poisson manifold and obtain the expression for the trace functional. The above ingredients are used to formulate a nonrelativistic quantum mechanics on noncommutative spaces of general form. All considered constructions are obtained as a formal series in the parameter of noncommutativity. In particular, the complete algebra of commutation relations between coordinates and conjugated momenta is a deformation of the standard Heisenberg algebra. As examples we consider a free particle and an isotropic harmonic oscillator on the rotational invariant noncommutative space.

Cohomology of Hom-Lie superalgebras and q-deformed Witt superalgebra

Hom-Lie algebra (superalgebra) structure appeared naturally in q-deformations, based on σ-derivations of Witt and Virasoro algebras (superalgebras). They are a twisted version of Lie algebras (superalgebras), obtained by deforming the Jacobi identity by a homomorphism. In this paper, we discuss the concept of α k -derivation, a representation theory, and provide a cohomology complex of Hom-Lie superalgebras. Moreover, we study central extensions. As application, we compute derivations and the second cohomology group of a twisted osp(1, 2) superalgebra and q-deformed Witt superalgebra.

Casimir invariants and the Jacobi identity in Dirac's theory of constrained Hamiltonian systems

We consider constrained Hamiltonian systems in the framework of Dirac's theory. We show that the Jacobi identity results from imposing that the constraints are Casimir invariants, regardless of whether the matrix of Poisson brackets between constraints is invertible or not. We point out that the proof we provide ensures the validity of the Jacobi identity everywhere in the phase space, and not just on the surface defined by the constraints. Two examples are considered: a finite-dimensional system with an odd number of constraints, and the Vlasov–Poisson reduction from Vlasov–Maxwell equations.

CATEGORICAL ASPECTS OF VIRTUALITY AND SELF-DISTRIBUTIVITY

This paper revolves around two main results. First, we propose a "hom-set" type categorification of virtual braid groups and positive virtual braid monoids in terms of "locally" braided objects in a symmetric category (SC). This "double braiding" approach provides a rich source of representations, and offers a natural categorical interpretation for virtual racks and for the twisted Burau representation. Second, we define self-distributive (SD) structures in an arbitrary SC. SD structures are shown to produce braided objects in a SC. As for examples, we interpret the associativity and the Jacobi identity in a SC as generalized self-distributivity, thus endowing associative and Leibniz algebras with a (pre-)braiding. A homology theory of categorical SD structures is developed using the "braided" techniques from [Lebed, Homologies of algebraic structures via braidings and quantum shuffles, to appear in J. Algebra], generalizing rack, bar, Leibniz and other familiar complexes.

The Hamiltonian Structures of μ-Equations Related to Periodic Peakons

The hierarchies of the μ-Camassa—Holm, two-component μ-Camassa—Holm and μ-modified Camassa—Holm equations are constructed from bi-Hamiltonian structures of the Korteweg-de Vries equation, the Ito equation and the modified Korteweg-de Vries equation. The key Hamiltonian operator that includes μ is ∂xμ — ∂3x, and the inner product used to define the Jacobi identity of the Hamiltonian operators is defined on the unit circle S1.

Integrand oxidation and one-loop colour-dual numerators in $ \mathcal{N}=4 $ gauge theory

We present a systematic method to determine BCJ numerators for one-loop amplitudes that explores the global constraints on the loop momentum dependence. We apply this method to amplitudes in N=4 gauge theory, working out detailed examples up to seven points in both the MHV and the NMHV sectors. The structure of Jacobi identities between BCJ numerators is seen to be closely connected to that of algebraic integrand reductions. We discuss the consequences for one-loop N=8 supergravity amplitudes obtained through the double copy prescription. Moreover, in the MHV sector, we show how to obtain simple BCJ box numerators using a conjectured relationship to amplitudes in self-dual gauge theory. We also introduce simpler trace-type formulas for integrand reductions.

Virtual color-kinematics duality: 6-pt 1-loop MHV amplitudes

We study 1-loop MHV amplitudes in N=4 super Yang-Mills theory and in N=8 supergravity. For Yang-Mills we find that the simple form for the full amplitude presented by Del Duca, Dixon and Maltoni naturally leads to one that has physical residues on all compact contours. After expanding the simple form in terms of standard scalar integrals, we introduce redundancies under certain symmetry considerations to impose the color-kinematics duality of Bern, Carrasco and Johansson (BCJ). For five particles we directly find the results of Carrasco and Johansson as well as a new compact form for the supergravity amplitude. For six particles we find that all kinematic dual Jacobi identities are encapsulated in a single functional equation relating the expansion coefficients. By the BCJ double-copy construction we obtain a formula for the corresponding N=8 supergravity amplitude. Quite surprisingly, all physical information becomes independent of the expansion coefficients modulo the functional equation. In other words, there is no need to solve the functional equation at all. This is quite welcome as the functional equation we find, using our restricted set of redundancies, actually has no solutions. For this reason we call these results virtual color-kinematics duality. We end with speculations about the meaning of an interesting global vs. local feature of the functional equation and the situation at higher points.

Meromorphic open-string vertex algebras

A notion of meromorphic open-string vertex algebra is introduced. A meromorphic open-string vertex algebra is an open-string vertex algebra in the sense of Kong and the author satisfying additional rationality (or meromorphicity) conditions for vertex operators. The vertex operator map for a meromorphic open-string vertex algebra satisfies rationality and associativity but in general does not satisfy the Jacobi identity, commutativity, the commutator formula, the skew-symmetry or even the associator formula. Given a vector space \documentclass[12pt]{minimal}\begin{document}$\mathfrak {h}$\end{document}h, we construct a meromorphic open-string vertex algebra structure on the tensor algebra of the negative part of the affinization of \documentclass[12pt]{minimal}\begin{document}$\mathfrak {h}$\end{document}h such that the vertex algebra structure on the symmetric algebra of the negative part of the Heisenberg algebra associated to \documentclass[12pt]{minimal}\begin{document}$\mathfrak {h}$\end{document}h is a quotient of this meromorphic open-string vertex algebra. We also introduce the notion of left module for a meromorphic open-string vertex algebra and construct left modules for the meromorphic open-string vertex algebra above.

Nonassociativity, Malcev algebras and string theory

AbstractNonassociative structures have appeared in the study of D‐branes in curved backgrounds. In recent work, string theory backgrounds involving three‐form fluxes, where such structures show up, have been studied in more detail. We point out that under certain assumptions these nonassociative structures coincide with nonassociative Malcev algebras which had appeared in the quantum mechanics of systems with non‐vanishing three‐cocycles, such as a point particle moving in the field of a magnetic charge. We generalize the corresponding Malcev algebras to include electric as well as magnetic charges. These structures find their classical counterpart in the theory of Poisson‐Malcev algebras and their generalizations. We also study their connection to Stueckelberg's generalized Poisson brackets that do not obey the Jacobi identity and point out that nonassociative string theory with a fundamental length corresponds to a realization of his goal to find a non‐linear extension of quantum mechanics with a fundamental length. Similar nonassociative structures are also known to appear in the cubic formulation of closed string field theory in terms of open string fields, leading us to conjecture a natural string‐field theoretic generalization of the AdS/CFT‐like (holographic) duality.

General Lower Bounds on Maximal Determinants of Binary Matrices

We give general lower bounds on the maximal determinant of $n \times n$ $\{+1,-1\}$-matrices, both with and without the assumption of the Hadamard conjecture. Our bounds improve on earlier results of de Launey and Levin (2010) and, for certain congruence classes of $n \bmod 4$, the results of Koukouvinos, Mitrouli and Seberry (2000). In an Appendix we give a new proof, using Jacobi's determinant identity, of a result of Szöllősi (2010) on minors of Hadamard matrices.

An algebraic approach to BCJ numerators

One important discovery in recent years is that the total amplitude of gauge theory can be written as BCJ form where kinematic numerators satisfy Jacobi identity. Although the existence of such kinematic numerators is no doubt, the simple and explicit construction is still an important problem. As a small step, in this note we provide an algebraic approach to construct these kinematic numerators. Under our Feynman-diagram-like construction, the Jacobi identity is manifestly satisfied. The corresponding color ordered amplitudes satisfy off-shell KK-relation and off-shell BCJ relation similar to the color ordered scalar theory. Using our construction, the dual DDM form is also established.

Consistently constrained SL(N) WZWN models and classical exchange algebra

Currents of the SL(N) WZWN model are constrained so that the remaining symmetry is a symmetry of constrained currents as well. Such consistency enables us to study the Poisson structure of constrained SL(N) WZWN models properly. We establish the Poisson brackets which satisfy the Jacobi identities owing to the classical Yang-Baxter equation. The Virasoro algebra is shown by using them. An SL(N) conformal primary is constructed. It satisfies a quadratic algebra, which might become an exchange algebra by its quantum deformation.

AXIOMATIC DIFFERENTIAL GEOMETRY II-3 -- ITS DEVELOPMENTS -- CHAPTER 3: THE GENERAL JACOBI IDENTITY

As the fourth paper of our series of papers concerned with ax- iomatic differential geometry, this paper is devoted to the general Jacobi iden- tity supporting the Jacobi identity of vector fields. The general Jacobi identity can be regarded as one of the few fundamental results belonging properly to smootheology.