# Lie Algebra

In mathematics, a Lie algebra ( /ˈliː/, not /ˈlaɪ/) is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" (after Sophus Lie) was introduced by Hermann Weyl in the 1930s. In older texts, the name "infinitesimal group" is used.

### Other articles related to "lie algebra, lie, algebra, lie algebras":

Lie Algebra - Category Theoretic Definition
... Using the language of category theory, a Lie algebra can be defined as an object A in Veck, the category of vector spaces over a field k of characteristic not 2, together with a morphism A ⊗ A → A, where ...
Special Linear Lie Algebra
... In mathematics, the special linear Lie algebra of order n (denoted ) is the Lie algebra of matrices with trace zero and with the Lie bracket ... This algebra is well studied and understood, and is often used as a model for the study of other Lie algebras ... The Lie group that it generates is the special linear group ...
Lie Coalgebra - The Lie Algebra On The Dual
... A Lie algebra structure on a vector space is a map which is skew-symmetric, and satisfies the Jacobi identity ... Dually, a Lie coalgebra structure on a vector space E is a linear map which is antisymmetric (this means that it satisfies, where is the canonical flip ) and satisfies ... The dual of the Lie bracket of a Lie algebra yields a map (the cocommutator) where the isomorphism holds in finite dimension dually for the dual of Lie comultiplication ...
Semisimple Lie Algebra - Properties - Linear
... The adjoint representation is injective, and so a semisimple Lie algebra is also a linear Lie algebra under the adjoint representation ... This may lead to some ambiguity, as every Lie algebra is already linear with respect to some other vector space (Ado's theorem), although not necessarily via the adjoint representation ...
Symplectic Representation
... theory, a symplectic representation is a representation of a group or a Lie algebra on a symplectic vector space (V, ω) which preserves the symplectic form ω ... ω if for all g in G and v, w in V, whereas a representation of a Lie algebra g preserves ω if for all ξ in g and v, w in V ... Thus a representation of G or g is equivalently a group or Lie algebra homomorphism from G or g to the symplectic group Sp(V,ω) or its Lie algebra ...

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