Proving the Jacobi Identity the Hard Way

Abstract

Abstract. Vector fields on smooth manifolds may be regarded as derivations of the algebra of smooth functions, as infinitesimal generators of flows, or as sections of the tangent bundle. The last point of view leads to a formula for the bracket which is not used very often and in terms of which such a basic matter as proving the Jacobi identity seems difficult. We present a conceptually simple proof of the Jacobi identity in terms of this formulation.Mathematics Subject Classification (2010). Primary 58A99; Secondary 18D05. Keywords. Vector fields, Jacobi identity, double vector bundles, triple vector bundles.There are three global formulas by which the bracket of vector fields can be calculated. Usually one interprets vector fields as derivations on the algebra of smooth functions, and the bracket is then the commutator of derivations. There is also the flow formula in which the bracket of vector fields is regarded as the Lie derivative of one field by the other.Thirdly, for vector fields , on a manifold , and ∈ , there is:Here :is the canonical involution. This formula involves some abuse: the RHS is a vertical vector in ( ) ( ) and therefore can be identified with an element of . For convenience we refer to (1) as the 'section formula' for the bracket.Formula (1) is much less widely used than the other two; one place in which it appears is [1, p. 297]. A proof can be extracted from [2, §3.4].By using derivations the proof of the Jacobi identity can be done in one line; a few moments experimentation with (1) may leave the reader with the impression that using the section formula is unnatural and unwieldy. The purpose of this paper is to show that there is a diagrammatic proof, very easy to visualize, starting from (1), using double and triple vector bundles. This will be important in work elsewhere -since (1) uses only the tangent functor and the canonical involution , it can be formulated in more abstract settings.

Warps, grids and curvature in triple vector bundles

“…In this section we consider the triple tangent bundle T 3 M of a manifold M and construct a grid on it, for which the Jacobi identity emerges as a consequence of the warp theorem. A version of this approach was given by Mackenzie [12]. We present here a clearer and more detailed calculation.…”

“…In a previous paper [12] one of us used a grid in the triple vector bundle T 3 M to express the Jacobi identity as a statement about the warps of the grids in the constituent double vector bundles. The proof given in that paper relied on a decomposition of T 3 M into seven copies of T M , and it was not clear whether the apparatus of grids and warps had provided a proof of the Jacobi identity or merely a formulation of it.…”

Warps and duality for double vector bundles

“…Warps and grids on double and triple vector bundles were introduced in [4], as a tool to formulate and prove the Jacobi identity of vector fields diagrammatically. These tools were further developed in [1] and in [2].…”

Higher-dimensional general Jacobi identities I

“…Now we consider the following four-dimensional analogue of the Jacobi identity. The principal objective in this paper is to establish a four-dimensional version of the general Jacobi identity underpinning the above identity (5). In a subsequent paper we will discuss a slew of higher-dimensional general Jacobi identities underlying the higher-dimensional Jacobi identities discussed in [1] and [12] (the former called them generalized Jacobi identities) from a coherent standpoint.…”

“…, (2,8) , (3,8) , (2,9) , (4,9) , (3, 10) , (4, 10) , (5, 6) , (5, 7) , (5,8) , (5,9) , (6, 7) , (6,8) , (6, 10) , (7,9) , (7,10) , (8,9) , (8, 10) , (9, 10) , (1, i 11,15 ) , (2, i 11,15 ) , (3, i 11,15 ) , (5, i 11,15 ) , (6, i 11,15 ) , (7, i 11,15 ) , (8, i 11,15 ) , (9, i 11,15 ) , (10, i 11,15…”