Generalised Lie Derivative: Theory & Applications

• Generalised Lie derivative is an extension of the classical Lie derivative incorporating minus-one-forms, redefining differential forms and vector fields for enriched geometry.
• It systematically integrates traditional operations—exterior derivative, inner product, and Lie bracket—into a framework that accounts for additional m-dependent corrections and global topological features.
• The framework supports advanced applications such as generalized Hamiltonian dynamics, affine connections, and metric extensions, ensuring a closed algebraic structure and global consistency.

The generalised Lie derivative is a formal extension of the traditional Lie derivative, designed to operate consistently on broader geometric and algebraic structures—specifically, on spaces combining ordinary differential forms with novel “minus-one-form” components, as well as on vector fields and connections valued in these generalised forms. The theory generalises essential operations of differential geometry—including the Lie derivative, inner product, Lie bracket, and the calculus of connections/curvature—to a setting where forms are systematically augmented, yielding rich algebraic structures with new geometric and physical implications.

1. Definition of the Generalised Lie Derivative for Forms and Vector Fields

Let $a = \alpha + \beta m$ denote a generalised $p$-form, where $\alpha$ is an ordinary $p$-form, $\beta$ a $(p+1)$-form, and $m$ the distinguished minus-one-form ($d m = \epsilon$, with constant $\epsilon$). For an ordinary vector field $v$, the generalised Lie derivative is modelled on the Cartan formula but adapted to this structure: $L_v a = i_v (d a) + d (i_v a)$ where the exterior derivative and contraction generalise as: \begin{align*} d a &= d\alpha + [d\beta + (-1)^{p+1}\epsilon \alpha] m, \ i_v a &= i_v \alpha + (i_v \beta) m, \end{align*} yielding (see Eq. (10) in the paper): $$L_v a = (i_v d\alpha + d (i_v \alpha)) + \left(i_v [d\beta + (-1)^{p+1}\epsilon\alpha] + d (i_v \beta) \right) m.$$ This derivative acts component-wise: $L_v a = L_v \alpha + (L_v \beta) m,$ so the operator distributes over the ordinary and minus-one components (Eq. (11)).

For a generalised form–valued vector field $V = v + (\tilde{v} m)$ (Eq. (35)), with $v$ a vector field and $\tilde{v}$ a $(1,1)$-tensor, the contraction and Lie derivative are defined as: \begin{align*} i_V r &= v^p (i_a r) \qquad (\text{Eq. (38)}), \ L_V r &= d(i_V r) + i_V (d r) \qquad (\text{Eq. (45)}), \end{align*} which again splits into explicit $m$-dependent terms (Eq. (47)). These preserve the graded Leibniz rule and reduce to standard expressions on ordinary forms and vectors. For vector fields of previous generalised type ([13], [14]), this is recovered by taking $\tilde{v}_\alpha^\beta = v^\beta dx^\alpha$ (Eq. (55)).

2. Structure of the Exterior Derivative and Global Consistency

The generalised exterior derivative is central for the consistency of the theory. It is defined on elements $a = \alpha + \beta m$ (Eq. (83)) as: $$d a = d\alpha + [d\beta + (-1)^{p+1}\epsilon \alpha] m, \qquad (\text{Eq. (86)})$$ with $\epsilon$ arising from $d m = \epsilon$. Internally, this derivative can be decomposed (Eq. (57)) as: $d = d^{(0)} + \epsilon d^{(1)},$ where $d^{(0)}$ is the ordinary exterior derivative and $d^{(1)}$ is a secondary operation reflecting the ambiguity in defining $d$ globally for $m$. This structure is non-trivial globally: the precise definition and patching of $m$ across manifold overlaps must satisfy compatibility rules (Eqs. (89)–(95)), ensuring that generalised forms and their derivatives are globally defined and consistent.

This decomposition is essential for a generalised Lie derivative that globally “remembers” the minus-one-form structure and correctly encodes the topological data of the manifold.

3. Generalised Lie Bracket and Closure Properties

The generalised Lie bracket for form–valued vector fields follows the prescription (Eq. (52)): $[L_V, L_W] r = L_{[V, W]} r,$ where $V$ and $W$ are generalised form–valued vector fields, $r$ is a generalised form, and $[V, W]$ is the corresponding generalised bracket. This ensures that the set of generalised vector fields—together with the generalised Lie derivative—forms a closed algebraic structure analogous to but richer than the ordinary Lie algebra of vector fields. The bracket and derivative extend all previously known formulas for generalised vector fields as a special case, maintaining the coherence and covariance necessary for further geometric constructions.

Expanding $d$ in terms of $d^{(0)}$ and $d^{(1)}$ allows explicit formulas for “pure” and “form–valued” parts. In particular, the operator

$$\hat{L}_V r = L_{v^{(0)}} r - [d^{(0)} i_{v^{(1)}} + i_{v^{(1)}} d^{(0)}] r$$

(Eq. (59)) encapsulates how body and form–valued components interact through the split exterior derivative.

4. Applications: Hamiltonian Vector Fields, Connections, and Metrics

Hamiltonian Generalised Vector Fields

The formalism naturally accommodates generalised Hamiltonian vector fields, which encode both standard symplectic geometry and new corrections. For a Hamiltonian generalised vector field $V_H$ associated with a generalised zero-form $H$, the defining equation is

$i_{V_H} \sigma = - dH,$

where $\sigma$ is a generalised symplectic two-form (Eq. (50)). When $\sigma$ and $H$ have both ordinary and $m$-components, the dynamics governed by $V_H$ generalise ordinary Hamilton’s equations, including new terms proportional to $\epsilon$ (representing, e.g., “damping” or “mass” corrections).

Generalised Affine Connections and Metrics

Generalised affine connections are constructed by promoting one-forms to generalised one-forms: $A = a + B m,$ with curvature

$F = dA + A \wedge A.$

The curvature $F$ contains not only the ordinary field strength $F_a$ but also additional $m$-dependent structure (Eq. (64)).

Generalised metrics are similarly promoted: $g = \Upsilon + X m,$ and the compatibility (zero non-metricity) leads to an extended version of the Levi–Civita connection and the Riemannian structure theorem (Eqs. (74), (80)–(82)). For $\epsilon = 0$, the standard Levi–Civita connection is recovered; $\epsilon \neq 0$ induces extra connection components driven by the generalised structure.

5. Overview and Significance

The generalised Lie derivative formalism unifies and extends several key ingredients of differential geometry:

• It provides a systematic way to define and compute derivatives on spaces enriched by minus-one-forms, capturing both the local and global topology of differential forms.
• The extended Cartan calculus incorporates m-dependent corrections and allows for new geometric interpretations and structures, such as generalised symplectic, metric, and connection data.
• The closure of the generalised Lie derivative and bracket under the algebra of form–valued vector fields ensures a robust framework suitable for exploring extensions in Hamiltonian dynamics, gauge theory, and global geometry.
• The role of the global structure of the exterior derivative, especially its decomposition and the gluing of m, is essential for ensuring well-defined operations on globally nontrivial manifolds.

The framework generalises all previously developed notions for generalised vector fields, rigorously encompasses both classical differential geometry and its generalisations, and paves the way for applications in areas such as generalised gauge theory, deformation quantisation, and general relativistic settings where standard geometry may be insufficient. The theory extends well beyond the local patch calculus by encoding the essential global and topological features inherently within the generalised calculus.