Jacobi Identity

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Hirofumi Tsumura - One of the best experts on this subject based on the ideXlab platform.

Cristiano Husu - One of the best experts on this subject based on the ideXlab platform.

James Lepowsky - One of the best experts on this subject based on the ideXlab platform.

  • on the concepts of intertwining operator and tensor product module in vertex operator algebra theory
    arXiv: Quantum Algebra, 2004
    Co-Authors: Yi-zhi Huang, James Lepowsky, Lin Zhang
    Abstract:

    We produce counterexamples to show that in the definition of the notion of intertwining operator for modules for a vertex operator algebra, the commutator formula cannot in general be used as a replacement axiom for the Jacobi Identity. We further give a sufficient condition for the commutator formula to imply the Jacobi Identity in this definition. Using these results we illuminate the crucial role of the condition called the ``compatibility condition'' in the construction of the tensor product module in vertex operator algebra theory, as carried out in work of Huang and Lepowsky. In particular, we prove by means of suitable counterexamples that the compatibility condition was indeed needed in this theory.

  • Application of a "Jacobi Identity" for vertex operator algebras to zeta values and differential operators
    arXiv: Quantum Algebra, 1999
    Co-Authors: James Lepowsky
    Abstract:

    We explain how to use a certain new "Jacobi Identity" for vertex operator algebras, announced in a previous paper (math.QA/9909178), to interpret and generalize recent work of S. Bloch's relating values of the Riemann zeta function at negative integers with a certain Lie algebra of operators.

  • On the D -module and formal-variable approaches to vertex algebras
    Topics in Geometry, 1996
    Co-Authors: Yi-zhi Huang, James Lepowsky
    Abstract:

    In a program to formulate and develop two-dimensional conformal field theory in the framework of algebraic geometry, Beilinson and Drinfeld [BD] have recently given a notion of “chiral algebra” in terms of D-modules on algebraic curves. This definition consists of a “skew-symmetry” relation and a “Jacobi Identity” relation in a categorical setting, and it leads to the operator product expansion for holomorphic quantum fields in the spirit of two-dimensional conformal field theory, as expressed in [BPZ], Because this operator product expansion, properly formulated, is known to be essentially a variant of the main axiom, the “Jacobi Identity” [FLM], for vertex (operator) algebras ([Borc], [FLM]; see [FLM] for the proof), the chiral algebras of [BD] amount essentially to vertex algebras.

  • On the D-module and formal-variable approaches to vertex algebras
    arXiv: Quantum Algebra, 1996
    Co-Authors: Yi-zhi Huang, James Lepowsky
    Abstract:

    In a program to formulate and develop two-dimensional conformal field theory in the framework of algebraic geometry, Beilinson and Drinfeld have recently given a notion of ``chiral algebra'' in terms of D-modules on algebraic curves. This definition consists of a ``skew-symmetry'' relation and a ``Jacobi Identity'' relation in a categorical setting. In this paper, we show directly that these chiral algebras are essentially the same as vertex algebras without vacuum vector (and without grading), by establishing an equivalence between the skew-symmetry and Jacobi Identity relations of Beilinson-Drinfeld and the (similarly-named, but different) skew-symmetry and Jacobi Identity relations in the formal-variable approach to vertex operator algebra theory as formulated by Borcherds, Frenkel-Lepowsky-Meurman and Frenkel-Huang-Lepowsky.

  • A Jacobi Identity for relative untwisted vertex operators
    Generalized Vertex Algebras and Relative Vertex Operators, 1993
    Co-Authors: Chongying Dong, James Lepowsky
    Abstract:

    We are now in a position to state and prove our “generalized Jacobi Identity” for relative untwisted vertex operators. Even for h* = 0, this result generalizes the Jacobi Identity in [FLM3] (Theorems 8.8.9 and 8.8.23) by removing all integrality restrictions on the “inner products” of lattice elements. Since the proof is very similar to that of Theorem 8.6.1 of [FLM3], we shall omit some computations, referring the reader to that proof for more details.

Sebastian Mizera - One of the best experts on this subject based on the ideXlab platform.

Arthemy V Kiselev - One of the best experts on this subject based on the ideXlab platform.

  • open problems in the kontsevich graph construction of poisson bracket symmetries
    arXiv: Mathematical Physics, 2019
    Co-Authors: Arthemy V Kiselev
    Abstract:

    Poisson brackets admit infinitesimal symmetries which are encoded using oriented graphs; this construction is due to Kontsevich (1996). We formulate several open problems about combinatorial and topological properties of the graphs involved, about integrability and analytic properties of such symmetry flows (in particular, for known classes of Poisson brackets), and about cohomological, differential geometric, and quantum aspects of the theory. Keywords: Poisson bracket, Jacobi Identity, infinitesimal symmetry, affine manifold, Poisson cohomology, graph complex.

  • The right-hand side of the Jacobi Identity: to be naught or not to be?
    Journal of Physics: Conference Series, 2016
    Co-Authors: Arthemy V Kiselev
    Abstract:

    The geometric approach to iterated variations of local functionals -e.g., of the (master-)action functional - resulted in an extension of the deformation quantisation technique to the set-up of Poisson models of field theory. It also allowed of a rigorous proof for the main inter-relations between the Batalin-Vilkovisky (BV) Laplacian Δ and variational Schouten bracket [,]. The ad hoc use of these relations had been a known analytic difficulty in the BV- formalism for quantisation of gauge systems; now achieved, the proof does actually not require the assumption of graded-commutativity. Explained in our previous work, geometry's self- regularisation is rendered by Gel'fand's calculus of singular linear integral operators supported on the diagonal.We now illustrate that analytic technique by inspecting the validity mechanism for the graded Jacobi Identity which the variational Schouten bracket does satisfy (whence Δ2 = 0, i.e., the BV-Laplacian is a differential acting in the algebra of local functionals). By using one tuple of three variational multi-vectors twice, we contrast the new logic of iterated variations - when the right-hand side of Jacobi's Identity vanishes altogether - with the old method: interlacing its steps and stops, it could produce some non-zero representative of the trivial class in the top- degree horizontal cohomology. But we then show at once by an elementary counterexample why, in the frames of the old approach that did not rely on Gel'fand's calculus, the BV-Laplacian failed to be a graded derivation of the variational Schouten bracket.

  • the right hand side of the Jacobi Identity to be naught or not to be
    Journal of Physics: Conference Series, 2016
    Co-Authors: Arthemy V Kiselev
    Abstract:

    The geometric approach to iterated variations of local functionals -e.g., of the (master-)action functional - resulted in an extension of the deformation quantisation technique to the set-up of Poisson models of field theory. It also allowed of a rigorous proof for the main inter-relations between the Batalin-Vilkovisky (BV) Laplacian Δ and variational Schouten bracket [,]. The ad hoc use of these relations had been a known analytic difficulty in the BV- formalism for quantisation of gauge systems; now achieved, the proof does actually not require the assumption of graded-commutativity. Explained in our previous work, geometry's self- regularisation is rendered by Gel'fand's calculus of singular linear integral operators supported on the diagonal.We now illustrate that analytic technique by inspecting the validity mechanism for the graded Jacobi Identity which the variational Schouten bracket does satisfy (whence Δ2 = 0, i.e., the BV-Laplacian is a differential acting in the algebra of local functionals). By using one tuple of three variational multi-vectors twice, we contrast the new logic of iterated variations - when the right-hand side of Jacobi's Identity vanishes altogether - with the old method: interlacing its steps and stops, it could produce some non-zero representative of the trivial class in the top- degree horizontal cohomology. But we then show at once by an elementary counterexample why, in the frames of the old approach that did not rely on Gel'fand's calculus, the BV-Laplacian failed to be a graded derivation of the variational Schouten bracket.

  • the right hand side of the Jacobi Identity to be naught or not to be
    arXiv: Mathematical Physics, 2014
    Co-Authors: Arthemy V Kiselev
    Abstract:

    The geometric approach [1312.1262] to iterated variations of local functionals -- e.g., of the (master-)action functional -- resulted in an extension of the deformation quantisation technique to the set-up of Poisson models of field theory [IHES/M/15/13]. It also allowed of a rigorous proof ([1312.1262],[1210.0726]) for the main inter-relations between the Batalin-Vilkovisky (BV) Laplacian $\Delta$ and variational Schouten bracket. The ad hoc use of these relations had been a known analytic difficulty in the BV-formalism for quantisation of gauge systems; now achieved, the proof does actually not require the assumption of graded-commutativity [1210.0726]. Explained in our previous work, geometry's self-regularisation is rendered by Gel'fand's calculus of singular linear integral operators supported on the diagonal. We now illustrate that analytic technique by inspecting the validity mechanism [1312.4140] for the graded Jacobi Identity which the variational Schouten bracket does satisfy (whence $\Delta^2=0$, i.e., the BV-Laplacian is a differential acting in the algebra of local functionals). By using one tuple of three variational multi-vectors twice, we contrast the new logic of iterated variations -- when the right-hand side of Jacobi's Identity vanishes altogether -- with the old method: interlacing its steps and stops, it could produce some non-zero representative of the trivial class in the top-degree horizontal cohomology. But we then show at once by an elementary counterexample why, in the frames of the old approach that did not rely on Gel'fand's calculus, the BV-Laplacian failed to be a graded derivation of the variational Schouten bracket. Keywords: Variational multi-vectors, Schouten bracket, Jacobi Identity, Batalin-Vilkovisky Laplacian, symbolic computations. PACS: 02.40.-k, 11.10.-z; also 02.30.Ik, 02.30.Jr, 11.15.-q, 11.30.-j

  • The Jacobi Identity for graded-commutative variational Schouten bracket revisited
    Physics of Particles and Nuclei Letters, 2014
    Co-Authors: Arthemy V Kiselev
    Abstract:

    This short note contains an explicit proof of the Jacobi Identity for variational Schouten bracket in ℤ2-graded commutative setup; an extension of the reasoning and assertion to the noncommutative geometry of cyclic words (see [1]) is immediate. The reasoning refers to the product bundle geometry of iterated variations (see [2]); no ad hoc regularizations occur anywhere in this theory.