We construct the universal central extension of the Lie algebra of exact divergence-free vector fields, proving a conjecture by Claude Roger from 1995. The proof relies on the analysis of a Leibniz algebra that underlies these vector fields. As an application, we construct the universal central extension of the (infinite-dimensional) Lie group of exact divergence-free diffeomorphisms of a compact 3-dimensional manifold.

Motivated by asymptotic symmetry groups in general relativity, we consider projective unitary representations ρ¯ of the Lie group Diffc(M) of compactly supported diffeomorphisms of a smooth manifold M that satisfy a so-called generalized positive energy condition. In particular, this captures representations that are in a suitable sense compatible with a KMS state on the von Neumann algebra generated by ρ¯. We show that if M is connected and dim(M)>1, then any such representation is necessarily trivial on the identity component Diffc(M)0. As an intermediate step towards this result, we determine the continuous second Lie algebra cohomology Hct2(Xc(M),R) of the Lie algebra of compactly supported vector fields. This is subtly different from Gelfand–Fuks cohomology in view of the compact support condition.

Let Γ be a discrete subgroup of a unimodular locally compact group G. M. Caspers et al. [Local and multilinear noncommutative de Leeuw theorems, Math. Ann. 388 (2024) 4251–4305] showed that the L_p -norm of a Fourier multiplier m: G → C on Γ can be bounded locally by its L_p -norm on G, modulo a constant c(A) which depends on the support A of m|_Γ . In the context where G is a connected Lie group with Lie algebra g, we develop tools to find explicit bounds on c(A) . We show that the problem reduces to: (1) The adjoint representation of the semisimple quotient s = g/r of g by the radical r ⊆ g (which was handled in the paper of M. Caspers et al. cited above). (2) The action of s on a set of real irreducible representations that arise from quotients of the commutator series of r . In particular, we show that c(G) = 1 for unimodular connected solvable Lie groups.

Using a nonlinear version of the tautological bundle over Graßmannians, we construct a transgression map for differential characters from M to the nonlinear Graßmannian Gr^S(M) of submanifolds of M of a fixed type S. In particular, we obtain prequantum circle bundles of the nonlinear Graßmannian endowed with the Marsden–Weinstein symplectic form. The associated Kostant–Souriau prequantum extension yields central Lie group extensions of a group of volume-preserving diffeomorphisms integrating Lichnerowicz cocycles.

Let Γ<G be a discrete subgroup of a locally compact unimodular group G. Let m∈C _b(G) be a p-multiplier on G with 1≤p<∞ and let T _m:L _p(G^)→L _p(G^) be the corresponding Fourier multiplier. Similarly, let Tm| _Γ:L _p(Γ^)→L _p(Γ^) be the Fourier multiplier associated to the restriction m| _Γ of m to Γ. We show that (Formula presented.) for a specific constant 0≤c(U)≤1 that is defined for every U⊆Γ. The function c quantifies the failure of G to admit small almost Γ-invariant neighbourhoods and can be determined explicitly in concrete cases. In particular, c(Γ)=1 when G has small almost Γ-invariant neighbourhoods. Our result thus extends the de Leeuw restriction theorem from Caspers et al. (Forum Math Sigma 3(e21):59, 2015) as well as de Leeuw’s classical theorem (Ann Math 81(2):364–379, 1965). For real reductive Lie groups G we provide an explicit lower bound for c in terms of the maximal dimension d of a nilpotent orbit in the adjoint representation. We show that c(B _ρ ^G)≥ρ ^-d/4 where B _ρ ^G is the ball of g∈G with ‖Ad _g‖<ρ. We further prove several results for multilinear Fourier multipliers. Most significantly, we prove a multilinear de Leeuw restriction theorem for pairs Γ<G with c(Γ)=1. We also obtain multilinear versions of the lattice approximation theorem, the compactification theorem and the periodization theorem. Consequently, we are able to provide the first examples of bilinear multipliers on nonabelian groups.

We construct an L1-algebra on the truncated canonical homology complex of a symplectic manifold, which naturally projects to the universal central extension of the Lie algebra of Hamiltonian vector fields.

There are eight possible Pin groups that can be used to describe the transformation behavior of fermions under parity and time reversal. We show that only two of these are compatible with general relativity, in the sense that the configuration space of fermions coupled to gravity transforms appropriately under the space-time diffeomorphism group.

Generalised spin structures describe spinor fields that are coupled to both general relativity and gauge theory. We classify those generalised spin structures for which the corresponding fields admit an infinitesimal action of the space–time diffeomorphism group. This can be seen as a refinement of the classification of generalised spin structures by Avis and Isham (Commun Math Phys 72:103–118, 1980).