Lie group action

A Lie group action is a smooth group action of a Lie group on a smooth manifold. Differentiating the action gives an infinitesimal action of its Lie algebra by vector fields.

A Lie group action on $M$ is proper if $(g,x)\mapsto (gx,x)$ from $G\times M$ to $M\times M$ is a proper map. A free proper smooth action has a smooth quotient manifold, with the quotient map a submersion and principal bundle. Proper actions with stabilizers can have orbit-type strata and singular quotients. These qualifications are necessary before treating an orbit space as a smooth phase space.

A smooth action of the compact abelian Lie group ${\mathbb{R}}^n/(2\pi \mathbb{Z}{)}^n$. Commuting vector fields with a common periodic normalization give such an action. In action-angle variables, the torus acts by translation of the angles while leaving the actions fixed.

The special linear group acts on determinant-one real symmetric matrices by matrix congruence. Symmetry and determinant are preserved, and signatures are preserved by Sylvester's law of inertia. The determinant-one set is a regular level set of dimension $n(n+1)/2-1$, with tangent directions $C=C^T$ satisfying $\mathrm{tr}(B^{-1}C)=0$. Its infinitesimal left-action fields are $XB+B{X}^T$; converting them to a Lie algebra representation requires the infinitesimal left-action sign convention.

The coadjoint action of a Lie group on the dual space of its Lie algebra is $({\mathrm{Ad}}_g^{\ast }\eta )(X)=\eta ({\mathrm{Ad}}_{g^{-1}}X)$. The inverse in this formula makes it a left group action.

A coadjoint orbit is the orbit of $\ell \in {\mathfrak{g}}^{\ast }$ under ${\mathrm{Ad}}_g^{\ast }\ell$, defined by $⟨{\mathrm{Ad}}_g^{\ast }\ell ,\xi ⟩=⟨\ell ,{\mathrm{Ad}}_{g^{-1}}\xi ⟩$. For a connected Lie group these orbits are the symplectic leaves of its Lie-Poisson manifold. Their tangent vectors are infinitesimal coadjoint actions, and their natural two-form is the Kirillov–Kostant–Souriau symplectic form.

With $⟨{\mathrm{ad}}_{\xi }^{\ast }\ell ,\eta ⟩=-⟨\ell ,[\xi ,\eta ]⟩$, the displayed form on a coadjoint orbit is well-defined: changing a generator by an element of the stabilizer changes no pairing. It is nondegenerate on orbit tangents, and its closedness follows from the Jacobi identity. This sign agrees with the positive Lie-Poisson bracket when Hamiltonian vector fields act by $X_f(g)=\{g,f\}$. A negative Lie-Poisson convention reverses the orbit form as well.

For a Lie group action and $X$ in its Lie algebra, the fundamental vector field is $X_M(p)={\frac{d}{dt}|}_{t=0}\exp (tX)\cdot p$.

For a smooth left Lie group action, define the fundamental vector field by $X^{\#}(p)={\frac{d}{ds}|}_0\exp (sX)\cdot p$. With the Lie bracket of vector fields $[U,V]f=U(Vf)-V(Uf)$, this map is an anti-homomorphism. The map $X\mapsto v_X=-X^{\#}$ is a Lie algebra representation. A right action with the positive exponential has the opposite homomorphism convention, so specifying action side and bracket convention prevents a sign ambiguity.