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A vector space with an alternating bilinear bracket satisfying the Jacobi identity.
The alternating axiom implies antisymmetry over every field. The converse needs characteristic different from 2, so antisymmetry is not used as an unrestricted substitute.
wp-lie-algebranlab-lie-algebrasage-lie-algebraEach relation below ends at this concept.
This 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.
Add an alternating bilinear bracket satisfying Jacobi.
Equip an existing carrier or structured object with additional chosen data, when such compatible data exists. Use a construction junction when several independently meaningful inputs must coexist on the same carrier or interact compatibly.
wp-lie-algebrastacks-algebra-textEach relation below starts at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
The Lie algebra supplies gauge generators, structure constants, and the local form of gauge-field interactions.
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-qcdwp-physics-gauge_field_theorynlab-physics-gauge_field_theorysage-lie-algebraThis is an authored directed relation from the source endpoint to the target endpoint.
Supply a Lie algebra over the same field as the vector-space input.
Feed several structures into a construction junction and impose compatibility between them.
mathlib-lie-moduleThis is an authored directed relation from the source endpoint to the target endpoint.
Infinitesimal symmetry generators lie in a Lie algebra and enter Noether currents.
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-noether_currentnlab-physics-noether_currentsage-lie-algebra