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The following will only deal with the Lie algebra, the question about the Lie group is far beyond my capabilities. The symplectic structure is (I guess) the one coming from the musical isomorphism of …

Maybe an even more elementary argument than the one of Tobias: The continuity of all involved operators is easy: simply all differential operators with smooth coefficients between sections of vector b …

In addition to Nik Weaver's references, let me just sketch the proof which is in fact not very difficult: A construction of Kaplansky (Rings of operators, Thm 26) shows that if $\mathcal{A}$ is a $*$ …

I suppose you want the action to be transitive as your title suggests. In this case, a classical theorem of Mostow (in 1950 for surfaces, in 2005 in general) says that for a compact homogeneous space …

Non-Hausdorffness shows up in several contexts when dealing with Lie groupoids: the integration (Lie's 3rd Theorem) for Lie algebroids to Lie groupoids will typically produce a non-Hausdorff one, if i …

One well-known example is the case of bivector fiels $\pi$. Then $[\pi, \pi] = 0$ is equivalent to say that $\{f, g\} = \pi(df, dg)$ is a Poisson bracket, i.e. satisfies the Jacobi identity. In this …

Being a bit late for the party, here is nevertheless a small answer. In fact, there is a rather explicit way to compute adjoints of every differential operator (any order) between (smooth, compactly s …

In general, this topology is coarser than the original topology of the manifold, and, without further assumptions, strictly coarser. It coincides with the original one iff the Lorentz manifold is stro …

This is a more technical question but it seems that there is some confusion in the literature on the choice of curves used to define the causal relations in time-oriented Lorentz manifolds: the infini …

I guess you want the topological equivalence to preserve the bundle structure of $TM \longrightarrow M$ otherwise it becomes a bit arbitrary, right? In this case a non-zero vertical vector field will …

Denote by $p_M: M \times N \longrightarrow M$ and $p_N: M \times N \longrightarrow N$ the canonical projections. Then you get an induced bilinear map from $\Omega^i(M) \times \Omega^j(N) \longrightarr …

Depending on you sign convention, this goes as follows. First you denote the bundle projection by $pr\colon E^* \longrightarrow X$. For a section $s \in \Gamma^\infty(E)$ you have a linear function $J …

This should surely be well-known by I have not been able to find a good reference to the following question: Given a smooth fiber bundle $\pi\colon P \longrightarrow M$ over a smooth manifold $M$ with …

This is a sort of standard construction you can find in several places. I don't know where this was done first though... OK: first you can extend your horizontal lift from vector fields to all (symm …

Consider a smooth (non-compact) manifold $M$ with a vector field $X$. Then we know that there is a open neighbourhood $U \subseteq M \times \mathbb{R}$ of $M \times \{0\}$ such that on $U$ the flow $\ …