Infinitesimal generator of a Lie group action

Definition

Let $G$ act smoothly on $M$ from the left through a smooth Lie group action, and let $\mathfrak{g}$ be the Lie algebra of $G$ . For $\xi\in\mathfrak{g}$ , its infinitesimal generator is the vector field

\xi_M(p)=\left.\frac{d}{dt}\right|_{t=0}\exp(t\xi)\mathbin{\cdot}p.

The resulting linear map $\mathfrak{g}\to\mathfrak{X}(M)$ , $\xi\mapsto\xi_M$ , is called the infinitesimal action. With this formula for a left action, it is a Lie algebra antihomomorphism: $[\xi_M,\eta_M]=-[\xi,\eta]_M$ .

Flow and stabilizers

The vector field $\xi_M$ is complete, and its flow is

\operatorname{Fl}^{\xi_M}_t(p)=\exp(t\xi)\mathbin{\cdot}p.

At a point $p$ , the evaluation map $\xi\mapsto\xi_M(p)$ has kernel equal to the Lie algebra of the stabilizer $G_p$ . Its image is the tangent space to the orbit through $p$ . The generators transform equivariantly:

d(g\mathbin{\cdot})_p\bigl(\xi_M(p)\bigr)=(\operatorname{Ad}_g\xi)_M(g\mathbin{\cdot}p).

Left and right action conventions

For a right action, the same unsigned formula $p\mathbin{\cdot}\exp(t\xi)$ produces a Lie algebra homomorphism. For a left action, replacing $\exp(t\xi)$ by $\exp(-t\xi)$ also produces a homomorphism. Both conventions occur in geometry, so bracket signs in moment-map and equivariance formulas depend on the chosen definition; see Marsden and Ratiu, §9.1.

Examples

For the left action of $G$ on itself by left multiplication, $\xi_G$ is the right-invariant vector field with value $\xi$ at the identity, explaining the antihomomorphism sign. For a linear representation $G\to\operatorname{GL}(V)$ , the generator is $\xi_V(v)=d\rho(\xi)v$ .

References

J. M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer, 2013. DOI record. Relevant: Chapter 21.
J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, 2nd ed., Springer, 1999. DOI record. Relevant: §9.1.