Lie group action

In differential geometry, a Lie group action on a manifold M is a group action by a Lie group G on M that is a differentiable map; in particular, it is a continuous group action. Together with a Lie group action by G, M is called a G-manifold. The orbit types of G form a stratification of M and this can be used to understand the geometry of M.

Let $\sigma: G \times M \to M, (g, x) \to g \cdot x$ be a group action. It is a Lie group action if it is differentiable. Thus, in particular, the orbit map $\sigma_x : G \to M, g \cdot x$ is differentiable and one can compute its differential at the identity element of G:

$\mathfrak{g} \to T_x M$.

If X is in $\mathfrak{g}$, then its image under the above is a tangent vector at x and, varying x, one obtains a vector field on M; the minus of this vector field is called the fundamental vector field associated with X and is denoted by $X^\#$. (The "minus" ensures that $\mathfrak{g} \to \Gamma(TM)$ is a Lie algebra homomorphism.) The kernel of the map can be easily shown (cf. Lie correspondence) to be the Lie algebra $\mathfrak{g}_x$ of the stabilizer $G_x$ (which is closed and thus a Lie subgroup of G.)

Let $P \to M$ be a principal G-bundle. Since G has trivial stabilizers in P, for u in P, $a \mapsto a^\#_u: \mathfrak{g} \to T_u P$ is an isomorphism onto a subspace; this subspace is called the vertical subspace. A fundamental vector field on P is thus vertical.

In general, the orbit space $M/G$ does not admit a manifold structure since, for example, it may not be Hausdorff. However, if G is compact, then $M/G$ is Hausdorff and if, moreover, the action is free, then $M/G$ is a manifold (in fact, $M \to M/G$ is a principal G-bundle.)^[1] This is a consequence of the slice theorem. If the "free action" is relaxed to "finite stabilizer", one instead obtains an orbifold (or quotient stack.)

A substitute for the construction of the quotient is the Borel construction from algebraic topology: assume G is compact and let $EG$ denote the universal bundle, which we can assume to be a manifold since G is compact, and let G act on $EG \times M$ diagonally; the action is free since it is so on the first factor. Thus, one can form the quotient manifold $M_G = (EG \times M)/G$. The constriction in particular allows one to define the equivariant cohomology of M; namely, one sets

$H^*_G(M) = H^*_{\text{dr}}(M_G)$,

where the right-hand side denotes the de Rham cohomology, which makes sense since $M_G$ has a structure of manifold (thus there is the notion of differential forms.)

If G is compact, then any G-manifold admits an invariant metric; i.e., a Riemannian metric with respect to which G acts on M as isometries.

See also

• Hamiltonian group action
• Equivariant differential form

References

1. de Faria, Edson; de Melo, Welington (2010), Mathematical Aspects of Quantum Field Theory, Cambridge Studies in Advanced Mathematics 127, Cambridge University Press, p. 69, ISBN 9781139489805.

• Michele Audin, Torus actions on symplectic manifolds, Birkhauser, 2004