#lie-theory
8 concepts
Concepts
Lie Group
A group with smooth manifold structure. Group operations are differentiable, representing continuous symmetries.
Lie Algebra
The tangent space at the identity of a Lie group, equipped with Lie bracket. Encodes infinitesimal structure of the group.
Exponential Map
A map from Lie algebra to Lie group. Generates finite transformations from infinitesimal generators.
Classical Lie Groups
Fundamental Lie groups represented as matrices: GL(n), SL(n), O(n), SO(n), U(n), SU(n), Sp(n).
Adjoint Representation
A representation of a Lie group acting on its own Lie algebra. Essential for studying Lie algebra structure.
Killing Form
A symmetric bilinear form on a Lie algebra. Essential for determining semisimplicity and classification.
Root System
A finite set of vectors encoding the structure of semisimple Lie algebras. Classified by Dynkin diagrams.
Semisimple Lie Algebra
A Lie algebra with no solvable ideals. Decomposes as direct sum of simple Lie algebras; Killing form is non-degenerate.