On the Moser trick for Lie subalgebras and foliations

Abstract: Given a smooth deformation of a Lie subalgebra, we establish a necessary and sufficient condition for its smooth triviality and derive an analogous criterion for Lie ideals. We then give a direct proof of the Moser trick for foliations, which forms the basis for extending this result to general Lie subalgebroids.

• The paper provides cohomological criteria that characterize the triviality of smooth deformations in Lie subalgebras, ideals, and foliations.
• It employs the Chevalley–Eilenberg formalism and Bott connection to construct explicit deformation complexes and cocycles.
• The work recasts the Moser trick, yielding both infinitesimal and global insights that advance rigidity and stability analyses in differential geometry.

On the Moser Trick for Lie Subalgebras and Foliations

Overview and Motivation

This paper investigates the deformation theory of Lie subalgebras and foliations, with a sharp focus on cohomological criteria for deformation triviality and precise technical refinements of the Moser trick. The central aim is to establish necessary and sufficient conditions under which smooth deformations of Lie subalgebras, Lie ideals, and foliations are trivial, and to elucidate the precise mechanisms underlying these triviality results in terms of deformation cohomology. The analysis culminates in a canonical, intrinsic proof of the Moser trick for foliations, expressed in terms of their associated deformation complex. The treatment is rigorous and systematic, offering both infinitesimal and global perspectives and framing all results within the modern language of Lie algebroid theory.

Deformations of Lie Subalgebras and Lie Ideals

The paper begins by formalizing smooth deformations for Lie subalgebras, Lie algebra morphisms, and Lie ideals. The relevant deformation complexes are constructed using the Chevalley-Eilenberg (CE) formalism. Explicit cohomological cocycles are associated to each deformation, with the first derivative of the deformed structure playing a central role in the construction of these cocycles.

For Lie subalgebra deformations $({h}_t,\iota_t)$ within a Lie algebra ${g}$, the paper shows:

• The cochain $\pi_{\mathrm{can}}^t \circ \dot{\iota}_t$ in $C^{\bullet}_{\iota_t}({h}_t; {g}/\iota_t({h}_t))$ is a cocycle for each $t$.
• Triviality of the deformation, i.e., existence of a smooth family of automorphisms $\Psi_t$ with $\iota_t \circ \psi_t = \Psi_t \circ \iota$, is equivalent to the existence of a smooth family $(X_t)_{t\in I} \subset Z^1({g};{g})$ such that $$\pi_{\mathrm{can}}^t \circ \dot{\iota}_t = \pi_{\mathrm{can}}^t \circ X_t \circ \iota_t$$ for all $t$.

An analogous criterion is established for Lie ideals, with the cocycle ${g}$0 and a corresponding cohomological vanishing condition involving automorphisms and ${g}$1-cocycles.

Notably, the criteria are expressed in terms of smooth vanishing of deformation cocycles in the relevant cohomology, providing a direct route to triviality that bypasses global path-connectedness assumptions for the subspace of subalgebras or ideals. The necessity and sufficiency proofs are framed infinitesimally, using time-dependent vector fields and flows on Lie groups and their automorphism structures.

Cohomological Moser Trick and Infinitesimal Techniques

The paper provides an explicit infinitesimal analogue of the Moser trick, inspired by Cárdenas-Struchiner's results for Lie group homomorphisms and subgroups. The technical heart of the treatment is Proposition 2.1, which systematically analyzes infinitesimal deformations of Lie algebra morphisms and relates the deformation cocycle to triviality under inner automorphisms and general automorphisms. The proofs appeal to flows of time-dependent vector fields, leveraging the equivalence between infinitesimal and global triviality for sufficiently small deformations.

These results not only unify earlier approaches, such as those of Richardson and Crainic-Schätz-Struchiner, but also extend them by providing direct cohomological criteria which are immune to the subtleties of local path-connectedness in the space of subalgebras.

Deformation Cohomology and Foliations

The second half of the paper addresses deformations of foliations, recasting existing deformation theory in terms of the Bott connection and the associated deformation complex ${g}$2. The deformation cocycle associated to a smooth deformation ${g}$3 is shown to be ${g}$4 for ${g}$5, and it is proven to be a ${g}$6-cocycle.

The Moser theorem for foliations is then established in full generality:

• Triviality of a smooth deformation implies smooth vanishing of the deformation cocycle in cohomology.
• Conversely, for compact manifolds, vanishing of the deformation cohomology classes as ${g}$7 varies assures triviality, via construction of an isotopy integrating the associated time-dependent vector field.

The proof is formulated entirely in terms of the deformation complex and relies on preservation properties of the Bott connection, the structure of the normal bundle, and the action of isotopies on foliations. The treatment avoids reliance on quasi-isomorphisms in more general Lie algebroid deformation contexts, thus clarifying the precise mechanism underpinning foliation rigidity.

Implications and Theoretical Significance

This work delivers a streamlined, intrinsic characterization of triviality for deformations of Lie subalgebras, ideals, and foliations, using cohomological vanishing as a necessary and sufficient condition. The approach is robust to topological subtleties and applies uniformly to infinitesimal and global settings. The Moser trick is recast as a cohomological criterion, providing a template for further extensions to Lie algebroid and subalgebroid deformation theory.

In practice, these results impact rigidity and stability analyses for symmetry structures in differential geometry and mathematical physics, as well as computations of moduli spaces of foliations and infinitesimal symmetries. The formalism is well-suited for generalization to ${g}$8-structures and higher homotopy algebras, as indicated by the connections to prior work by Vitagliano, Huebschmann, and Ji on deformation complexes for foliations.

Further, the equivalence between gauge equivalence and geometric equivalence in deformation contexts, as well as the links to the theory of groupoid deformations, are highlighted for future exploration.

Conclusion

The paper establishes direct cohomological criteria for the triviality of smooth deformations of Lie subalgebras, Lie ideals, and foliations, presenting precise and formal development of the Moser trick in these settings (2606.25848). The results unify deformation-theoretic approaches, eliminate reliance on topological path-connectedness, and provide constructive infinitesimal methods applicable to broader classes of infinitesimal symmetries. Future applications include extensions to general Lie subalgebroid rigidity, refined moduli computations, and comparison of gauge versus geometric equivalence in deformation theory.