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Concept
Lie group action
Summary
In differential geometry, a Lie group action is a group action adapted to the smooth setting: G is a Lie group, M is a smooth manifold, and the action map is differentiable.
TOC
Let be a (left) group action of a Lie group G on a smooth manifold M; it is called a Lie group action (or smooth action) if the map is differentiable. Equivalently, a Lie group action of G on M consists of a Lie group homomorphism . A smooth manifold endowed with a Lie group action is also called a G-manifold.
The fact that the action map is smooth has a couple of immediate consequences:
the stabilizers of the group action are closed, thus are Lie subgroups of G
the orbits of the group action are immersed submanifolds.
Forgetting the smooth structure, a Lie group action is a particular case of a continuous group action.
For every Lie group G, the following are Lie group actions:
the trivial action of G on any manifold
the action of G on itself by left multiplication, right multiplication or conjugation
the action of any Lie subgroup on G by left multiplication, right multiplication or conjugation
the adjoint action of G on its Lie algebra .
Other examples of Lie group actions include:
the action of on M given by the flow of any complete vector field
the actions of the general linear group and of its Lie subgroups on by matrix multiplication
more generally, any Lie group representation on a vector space
any Hamiltonian group action on a symplectic manifold
the transitive action underlying any homogeneous space
more generally, the group action underlying any principal bundle
Following the spirit of the Lie group-Lie algebra correspondence, Lie group actions can also be studied from the infinitesimal point of view. Indeed, any Lie group action induces an infinitesimal Lie algebra action on M, i.e. a Lie algebra homomorphism . Intuitively, this is obtained by differentiating at the identity the Lie group homomorphism , and interpreting the set of vector fields as the Lie algebra of the (infinite-dimensional) Lie group .
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Lie groupoid
In mathematics, a Lie groupoid is a groupoid where the set of s and the set of morphisms are both manifolds, all the operations (source and target, composition, identity-assigning map and inversion) are smooth, and the source and target operations are submersions. A Lie groupoid can thus be thought of as a "many-object generalization" of a Lie group, just as a groupoid is a many-object generalization of a group. Accordingly, while Lie groups provide a natural model for (classical) continuous symmetries, Lie groupoids are often used as model for (and arise from) generalised, point-dependent symmetries.
Principal bundle
In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product of a space with a group . In the same way as with the Cartesian product, a principal bundle is equipped with An action of on , analogous to for a product space. A projection onto . For a product space, this is just the projection onto the first factor, . Unlike a product space, principal bundles lack a preferred choice of identity cross-section; they have no preferred analog of .