Lie group

A Lie group is a group that is also a smooth manifold, with smooth multiplication and inversion. It combines continuous geometric structure with algebraic symmetry.

Group: A set equipped with an associative operation, an identity element, and inverses for every element. Its algebraic axioms remain part of a Lie group’s definition.
Lie algebra: A vector space with a bilinear, antisymmetric bracket satisfying the Jacobi identity. It captures a Lie group’s infinitesimal structure at its identity.
General linear group: The group of invertible linear transformations of a finite-dimensional vector space. Its matrices form a basic Lie group under multiplication.
Noether's theorem: A theorem linking continuous symmetries of a physical action to conserved quantities. Lie group symmetries yield the conservation laws described by the theorem.
Discrete group: A group regarded as a discrete topological space, with no nontrivial local continuous structure. Unlike a Lie group, it has no positive-dimensional smooth directions.
Smooth manifold: A space locally modeled on Euclidean space, with smoothly compatible coordinate charts. It supplies the geometric structure underlying a Lie group.
Exponential map: A map from a Lie algebra to its Lie group that sends infinitesimal generators to one-parameter subgroups. It connects algebraic directions to continuous motions in the group.
Special orthogonal group: The group of real orthogonal matrices with determinant one. It represents rotations and is a central compact Lie group.
Lie symmetry analysis: A method for finding continuous transformation symmetries of differential equations. Its transformations are organized as local Lie groups.
Topological group: A group with a topology for which multiplication and inversion are continuous. It requires continuity but not the smooth manifold structure of a Lie group.

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