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Note that a "complex Lie group" is defined as a complex analytic manifold that is also a group whose multiplication and inversion are each given by a holomorphic map. The dimensions in the table below are dimensions over C. Note that every complex Lie group/algebra can also be viewed as a real Lie group/algebra of twice the dimension.
The orthogonal group has two connected components; the identity component is called the special orthogonal group (), consisting of the orthogonal matrices with determinant 1. Both groups have the same Lie algebra (), the subspace of skew-symmetric matrices in (,) (=).
The Lie algebra of a simple Lie group is a simple Lie algebra. This is a one-to-one correspondence between connected simple Lie groups with trivial center and simple Lie algebras of dimension greater than 1. (Authors differ on whether the one-dimensional Lie algebra should be counted as simple.)
That is, there may be a group corresponding to the Lie algebra, but it might not be nice enough to be called a Lie group, or the connection between the group and the Lie algebra might not be nice enough (for example, failure of the exponential map to be onto a neighborhood of the identity). It is the "nice enough" that is not universally defined.
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 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 ...
A nilpotent element of a semisimple Lie algebra [1] is an element x such that the adjoint endomorphism ... J.-P. Serre, "Lie algebras and Lie groups", Benjamin (1965 ...
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 ...