Lie group action

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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 σ:G×M→M,(g,x)↦g⋅xdisplaystyle sigma :Gtimes Mto M,(g,x)mapsto gcdot x be a group action. It is a Lie group action if it is differentiable. Thus, in particular, the orbit map σx:G→M,g↦g⋅xdisplaystyle sigma _x:Gto M,gmapsto gcdot x is differentiable and one can compute its differential at the identity element of G:

g→TxMdisplaystyle mathfrak gto T_xM.

If X is in gdisplaystyle 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#displaystyle X^#. (The "minus" ensures that g→Γ(TM)displaystyle mathfrak gto Gamma (TM) is a Lie algebra homomorphism.) The kernel of the map can be easily shown (cf. Lie correspondence) to be the Lie algebra gxdisplaystyle mathfrak g_x of the stabilizer Gxdisplaystyle G_x (which is closed and thus a Lie subgroup of G.)

Let P→Mdisplaystyle Pto M be a principal G-bundle. Since G has trivial stabilizers in P, for u in P, a↦au#:g→TuPdisplaystyle amapsto a_u^#:mathfrak gto T_uP 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/Gdisplaystyle M/G does not admit a manifold structure since, for example, it may not be Hausdorff. However, if G is compact, then M/Gdisplaystyle M/G is Hausdorff and if, moreover, the action is free, then M/Gdisplaystyle M/G is a manifold (in fact, M→M/Gdisplaystyle Mto 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 EGdisplaystyle EG denote the universal bundle, which we can assume to be a manifold since G is compact, and let G act on EG×Mdisplaystyle EGtimes M diagonally; the action is free since it is so on the first factor. Thus, one can form the quotient manifold MG=(EG×M)/Gdisplaystyle M_G=(EGtimes M)/G. The constriction in particular allows one to define the equivariant cohomology of M; namely, one sets

HG∗(M)=Hdr∗(MG)displaystyle H_G^*(M)=H_textdr^*(M_G),

where the right-hand side denotes the de Rham cohomology, which makes sense since MGdisplaystyle 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