Lie Theory

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10 domain-specific abstractions whose origin domain is Lie Theory.

• Abelian Lie group — A smooth Lie group whose multiplication is commutative, combining a finite-dimensional manifold with an abelian group structure and smooth operations.
• En (Lie algebra) — The Lie or Kac–Moody algebra family associated with the E-n branching Dynkin diagram, including exceptional finite cases and indefinite extensions.
• Exponential map (Lie theory) — The canonical smooth map sending a Lie-algebra element to the time-one point of its one-parameter subgroup in the Lie group.
• Infinitesimal transformation — A first-order generator describing the tangent direction of a continuous one-parameter family of transformations at the identity.
• Iwasawa decomposition — A factorization G=KAN of a connected real semisimple Lie group into maximal compact, abelian and nilpotent subgroups, generalizing matrix QR decomposition.
• Killing form — Pair two elements of a finite-dimensional Lie algebra by tracing the composition of their adjoint endomorphisms, obtaining a canonical symmetric invariant bilinear form whose degeneracy diagnoses structure.
• Loop Group — A group of maps from a circle into a Lie group under pointwise multiplication, often equipped with smoothness, based-loop, and central-extension structure.
• One-parameter group — A continuous homomorphism from the additive real numbers into a topological group, representing a continuously parameterized group action or flow.
• Real form (Lie theory) — A real Lie algebra or group whose scalar extension to the complex numbers recovers a specified complex Lie algebra or group.
• Restricted root system — The root system obtained by restricting a semisimple Lie algebra’s roots to a maximal abelian subspace in the noncompact part of a symmetric decomposition.