Lie group

Target entity: Lie group
Context triple: [Sophus Lie, hasConceptNamedAfter, Lie group]

• A. Lie theory
  Lie theory is a branch of mathematics that studies continuous symmetry through Lie groups and Lie algebras, with deep applications in geometry, analysis, and theoretical physics.
• B. Lie pseudogroup
  A Lie pseudogroup is a collection of local diffeomorphisms on a manifold that is closed under composition, inversion, and restriction, generalizing the concept of a Lie group to transformations defined only locally.
• C. Lie subgroup
  A Lie subgroup is a subgroup of a Lie group that is itself a Lie group and an embedded submanifold, inheriting compatible smooth and group structures from the ambient Lie group.
• D. Lorentz group
  The Lorentz group is the mathematical group of spacetime symmetries in special relativity, consisting of all rotations and boosts that preserve the Minkowski spacetime interval.
• E. Euclidean group
  The Euclidean group is the group of all distance-preserving transformations of Euclidean space, consisting of rotations, reflections, and translations.
• F. None of above. chosen
• G. Unsure - the case is ambiguous/there is not enough information to decide.

Target entity: Lie group
Target entity description: A Lie group is a mathematical structure that is both a smooth manifold and a group, where the group operations are differentiable and used to study continuous symmetries.

• A. Lie theory
  Lie theory is a branch of mathematics that studies continuous symmetry through Lie groups and Lie algebras, with deep applications in geometry, analysis, and theoretical physics.
• B. Lie pseudogroup
  A Lie pseudogroup is a collection of local diffeomorphisms on a manifold that is closed under composition, inversion, and restriction, generalizing the concept of a Lie group to transformations defined only locally.
• C. Lie subgroup
  A Lie subgroup is a subgroup of a Lie group that is itself a Lie group and an embedded submanifold, inheriting compatible smooth and group structures from the ambient Lie group.
• D. Lorentz group
  The Lorentz group is the mathematical group of spacetime symmetries in special relativity, consisting of all rotations and boosts that preserve the Minkowski spacetime interval.
• E. Euclidean group
  The Euclidean group is the group of all distance-preserving transformations of Euclidean space, consisting of rotations, reflections, and translations.
• F. None of above. chosen