when can we lift an action of Lie algebra? - MathOverflow

when can we lift an action of Lie algebra?

2011-08-24T13:36:25Z

Suppose $G$ is a Lie group, $\mathfrak{g}$ its Lie algebra, if we have a smooth representation $(\pi,V)$, then it induces an action of $\mathfrak{g}$ on $V$. Now conversely, if we have a nice (with properties you may assume) action of $\mathfrak{g}$ on $V$, can we say such action arises from some unique smooth action of $G$?

Here we may assume $G$ to be simply connected if needed. Thank you.

Answer by Giuseppe for when can we lift an action of Lie algebra?

2011-08-24T14:46:27Z

Let be $G$ a simply connected Lie group, $\mathfrak{g}$ its Lie algebra and $M$ an arbitrary smooth manifold. Let be $\zeta$ a smooth action of $\mathfrak{g}$ on a $M$, i.e. $\zeta:X\in\mathfrak{g}\to\mathfrak{X}(M)$ is a Lie algebra homomorphism.

Then there exists a local left action $\Phi$ of $G$ on $M$ such that, for any $X\in\mathfrak{g}$, the t-time local flow of $\zeta(X)$ is given by $m\mapsto\Phi(e^{-t.X},m)$

In general the action of $\mathfrak{g}$ on $M$ can only be lifted to a local left action $\Phi$ of $G$ on $M$, i.e. defined only on a neighborhood of $\{e\}\times M$ in $G\times M$.

But, if $\zeta(X)$ is a complete vector on $M$ for any $X\in\mathfrak{g}$, then $\zeta$ can be lifted to a global left action of $G$ on $M$.

These results should be found in the work of Richard Palais on the Lie theory of transformation groups.

Answer by Eric O. Korman for when can we lift an action of Lie algebra?

2011-08-24T14:58:04Z

Yes if $G$ is connected and simply connected, since in that case there is a one to one correspondence between Lie group homomorphisms $G\to H$ and Lie algebra homomorphisms $\mathfrak g \to \mathfrak h$. Since a representation of $\mathfrak g$ is just a Lie algebra homomorphism $\mathfrak g \to \mathfrak{gl}(V)$, your desired result follows.

EDIT: I'm assuming $V$ is finite dimensional.

Answer by Salvatore Siciliano for when can we lift an action of Lie algebra?

2011-08-24T16:14:02Z

For a reference about the well-known fact that finite-dimensional representations of a connected and simply connected Lie group are in one-to-one correnspondence with finite-dimensional representations of its Lie algebra, the OP is referred e.g. to "Fulton-Harris: Representation Theory", Section 8.1.

Answer by Alain Valette for when can we lift an action of Lie algebra?

2011-08-25T05:11:35Z

Let $V$ be the space of smooth, compactly supported functions on $\mathbb{R}$, which vanish at $0$ together with all their derivatives. Define $X:V\rightarrow V$ by $Xf=f'$. Then $X$ defines an action of the Lie algebra of $\mathbb{R}$ on $V$, which does not integrate to an action of $\mathbb{R}$.

Answer by Joseph Wolf for when can we lift an action of Lie algebra?

2011-10-25T18:51:29Z

Let $\pi$ represent a finite dimensional real Lie algebra $\mathfrak g$ on a Hilbert space $\mathcal H$ by skew-adjoint operators. Then $\pi$ integrates to the connected simply connected Lie group $G$ with Lie algebra $\mathfrak g$ if, and only if, the elements of $\pi(\mathfrak g)$ have a common invariant dense domain. This is an old result of Moshe Flato, Daniel Sternheimer and others. My apologies to the mathematical physicists whose names I have omitted.