Related papers: Higher-Dimensional general Jacobi identities I
As the fourth paper of our series of papers concerned with axiomatic differential geometry, this paper is devoted to the general Jacobi identity supporting the Jacobi identity of vector fields. The general Jacobi identity can be regarded as…
We first present some identities involving the Pochhammer symbol (rising factorial). We also recall and present some new properties of the Jacobi polynomials. We use them to expand a general hypergeometric function in an orthogonal series…
The Jacobi identity is the key relation in the definition of a Lie algebra. In the last decade, it also appeared at the heart of the theory of finite type invariants of knots, links and 3-manifolds (and is there called the IHX-relation). In…
On the basis of the generalizations of the Jacobi identity found by the author some identities satisfied by the curvature and torsion of a covariant differentiation are derived. A kind of the generalized covariant differentiation is…
In our previous paper (Axiomatic Differential Geometry II-3) we have discussed the general Jacobi identity, from which the Jacobi identity of vector fields follows readily. In this paper we derive Jacobi-like identities of…
Vertex algebras in higher dimensions provide an algebraic framework for investigating axiomatic quantum field theory with global conformal invariance. We develop further the theory of such vertex algebras by introducing formal calculus…
A new family of $n$-dimensional solutions of the Jacobi identities is characterized. Such a family is very general, thus unifying in a common framework many different well-known Poisson systems seemingly unrelated. This unification is not…
The Jacobi identity is one of the properties that are used to define the concept of Lie algebra and in this context is closely related to associativity. In this paper we provide a complete description of all bivariate polynomials that…
Frolicher and Nijenhuis recognized well in the middle of the previous century that the Lie bracket and its Jacobi identity could and should exist beyond Lie algebras. Nevertheless the conceptual meaning of their discovery has been obscured…
By definition the identities $[x_1, x_2] + [x_2, x_1] = 0$ and $[x_1, x_2, x_3] + [x_2, x_3, x_1] + [x_3, x_1, x_2] = 0$ hold in any Lie algebra. It is easy to check that the identity $[x_1, x_2, x_3, x_4] + [x_2, x_1, x_4, x_3] + [x_3,…
In this paper we generalize the famous Jacobi's triple product identity, considered as an identity for theta functions with characteristics and their derivatives, to higher genus/dimension. By applying the results and methods developed in…
A methodical analysis of the research related to the article, ``Sur les groupes continus'', of Henri Poincar\'{e} reveals many historical misconceptions and inaccuracies regarding his contribution to Lie theory. A thorough reading of this…
Just as the Jacobi identity of vector fields is a natural consequence of the general Jacobi identity of microcubes in synthetic differential geometry, it is to be shown in this paper that the graded Jacobi identity of the…
We introduce two remarkable identities written in terms of single commutators and anticommutators for any three elements of arbitrary associative algebra. One is a consequence of other (fundamental identity). From the fundamental identity,…
We describe the definition of Jacobi (generalized)-Lie bialgebras $(({\bf{g}},\phi_{0}),({\bf{g}}^{*},X_{0}))$ in terms of structure constants of the Lie algebras ${\bf{g}}$ and ${\bf{g}}^{*}$ and components of their 1-cocycles $X_{0}\in…
We prove a generalized rationality property and a new identity that we call the ``Jacobi identity'' for intertwining operator algebras. Most of the main properties of genus-zero conformal field theories, including the main properties of…
A set of generalized superalgebras containing arbitrary tensor p-form operators is considered in dimensions $D=2n+1$ for $n=1,4 mod 4$ and the general conditions for its existence expressed in the form of generalized Jacobi identities is…
In this survey, we review the classical Hamilton Jacobi theory from a geometric point of view in different geometric backgrounds. We propose a Hamilton Jacobi equation for different geometric structures attending to one particular…
Using general identities for difference operators, as well as a technique of symbolic computation and tools from probability theory, we derive very general kth order (k \ge 2) convolution identities for Bernoulli and Euler polynomials. This…
We prove that the Jacobi identity for the generalized Poisson bracket is satisfied in the generalization of Heisenberg picture quantum mechanics recently proposed by one of us (SLA). The identity holds for any combination of fermionic and…