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A smooth manifold whose group operations are smooth.
wp-j-lie-groupnlab-lie-grouplee-smooth-manifoldsEach relation below ends at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
Require multiplication and inversion to be smooth.
Feed several structures into a construction junction and impose compatibility between them.
nlab-lie-grouplee-smooth-manifoldsEach relation below starts at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
A compact Lie group has compact underlying topology.
Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.
wp-compact_lie_groupnlab-compact_lie_groupThis is an authored directed relation from the source endpoint to the target endpoint.
Gauge theories use Lie groups to encode local symmetry transformations and classify gauge fields and matter representations.
A mathematical concept supplies part of the formal language, state space, representation, or analytic machinery used by a scientific or mathematical-physics concept. The source is the mathematical predecessor; this does not claim that physical content follows from mathematics alone.
cern-qft-intropdg-standard-modelwp-physics-gauge_field_theorynlab-physics-gauge_field_theorynlab-lie-groupThis is an authored directed relation from the source endpoint to the target endpoint.
Take the tangent space at the identity with the bracket induced by left-invariant vector fields.
Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.
nlab-lie-groupwp-lie-algebrastacks-algebra-textThis is an authored directed relation from the source endpoint to the target endpoint.
The Lorentz group is the Lie group preserving a Lorentzian quadratic form.
A more specific theory, model, entity class, or regime is obtained by restricting or extending the scope of a broader framework.
wp-lorentz_groupnlab-lorentz_groupThis is an authored directed relation from the source endpoint to the target endpoint.
Continuous physical symmetries are modeled by Lie groups.
A mathematical concept supplies part of the formal language, state space, representation, or analytic machinery used by a scientific or mathematical-physics concept. The source is the mathematical predecessor; this does not claim that physical content follows from mathematics alone.
wp-physics-physical_symmetrynlab-physics-symmetrynlab-lie-grouplee-smooth-manifolds