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Lie algebras were introduced to study the concept of infinitesimal transformations by Sophus Lie in the 1870s, [1] and independently discovered by Wilhelm Killing [2] in the 1880s. The name Lie algebra was given by Hermann Weyl in the 1930s; in older texts, the term infinitesimal group was used.
This article gives a table of some common Lie groups and their associated Lie algebras.. The following are noted: the topological properties of the group (dimension; connectedness; compactness; the nature of the fundamental group; and whether or not they are simply connected) as well as on their algebraic properties (abelian; simple; semisimple).
A Lie algebra representation also arises in nature. If : G → H is a homomorphism of (real or complex) Lie groups, and and are the Lie algebras of G and H respectively, then the differential: on tangent spaces at the identities is a Lie algebra homomorphism.
The foundation of Lie theory is the exponential map relating Lie algebras to Lie groups which is called the Lie group–Lie algebra correspondence. The subject is part of differential geometry since Lie groups are differentiable manifolds. Lie groups evolve out of the identity (1) and the tangent vectors to one-parameter subgroups generate the ...
Suppose G is a closed subgroup of GL(n;C), and thus a Lie group, by the closed subgroups theorem.Then the Lie algebra of G may be computed as [2] [3] = {(;)}. For example, one can use the criterion to establish the correspondence for classical compact groups (cf. the table in "compact Lie groups" below.)
The Lie algebra of any compact Lie group (very roughly: one for which the symmetries form a bounded set) can be decomposed as a direct sum of an abelian Lie algebra and some number of simple ones. The structure of an abelian Lie algebra is mathematically uninteresting (since the Lie bracket is identically zero); the interest is in the simple ...
abelian 1. An abelian Lie group is a Lie group that is an abelian group. 2. An abelian Lie algebra is a Lie algebra such that [,] = for every , in the algebra. adjoint 1. An adjoint representation of a Lie gro
A finite-dimensional simple complex Lie algebra is isomorphic to either of the following: , , (classical Lie algebras) or one of the five exceptional Lie algebras. [1]To each finite-dimensional complex semisimple Lie algebra, there exists a corresponding diagram (called the Dynkin diagram) where the nodes denote the simple roots, the nodes are jointed (or not jointed) by a number of lines ...