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In abstract algebra, specifically the theory of Lie algebras, Serre's theorem states: given a (finite reduced) root system, there exists a finite-dimensional semisimple Lie algebra whose root system is the given .
The simple Lie algebras are classified by the connected Dynkin diagrams. Every semisimple Lie algebra over an algebraically closed field of characteristic 0 is a direct sum of simple Lie algebras (by definition), and the finite-dimensional simple Lie algebras fall in four families – A n, B n, C n, and D n – with five exceptions E 6, E 7, E ...
The correspondence between Lie algebras and Lie groups is used in several ways, including in the classification of Lie groups and the representation theory of Lie groups. For finite-dimensional representations, there is an equivalence of categories between representations of a real Lie algebra and representations of the corresponding simply ...
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.
This is a glossary for the terminology applied in the mathematical theories of Lie groups and Lie algebras. For the topics in the representation theory of Lie groups and Lie algebras, see Glossary of representation theory. Because of the lack of other options, the glossary also includes some generalizations such as quantum group
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.)
In mathematics, a regular element of a Lie algebra or Lie group is an element whose centralizer has dimension as small as possible. For example, in a complex semisimple Lie algebra, an element is regular if its centralizer in has dimension equal to the rank of , which in turn equals the dimension of some Cartan subalgebra (note that in earlier papers, an element of a complex semisimple Lie ...