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

${\displaystyle {\mathfrak {g}}\to T_{x}M}$.

If X is in ${\displaystyle {\mathfrak {g}}}$, 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 ${\displaystyle X^{\#}}$. (The "minus" ensures that ${\displaystyle {\mathfrak {g}}\to \Gamma (TM)}$ is a Lie algebra homomorphism.) The kernel of the map can be easily shown (cf. Lie correspondence) to be the Lie algebra ${\displaystyle {\mathfrak {g}}_{x}}$ of the stabilizer ${\displaystyle G_{x}}$ (which is closed and thus a Lie subgroup of G.)

Let ${\displaystyle P\to M}$ be a principal G-bundle. Since G has trivial stabilizers in P, for u in P, ${\displaystyle a\mapsto a_{u}^{\#}:{\mathfrak {g}}\to T_{u}P}$ 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 ${\displaystyle M/G}$ does not admit a manifold structure since, for example, it may not be Hausdorff. However, if G is compact, then ${\displaystyle M/G}$ is Hausdorff and if, moreover, the action is free, then ${\displaystyle M/G}$ is a manifold (in fact, ${\displaystyle M\to M/G}$ 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 ${\displaystyle EG}$ denote the universal bundle, which we can assume to be a manifold since G is compact, and let G act on ${\displaystyle EG\times M}$ diagonally; the action is free since it is so on the first factor. Thus, one can form the quotient manifold ${\displaystyle M_{G}=(EG\times M)/G}$. The constriction in particular allows one to define the equivariant cohomology of M; namely, one sets

${\displaystyle H_{G}^{*}(M)=H_{\text{dr}}^{*}(M_{G})}$,

where the right-hand side denotes the de Rham cohomology, which makes sense since ${\displaystyle M_{G}}$ 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
• isotropy representation

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