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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 third theorem on the list stated the Jacobi identity for the infinitesimal transformations of a local Lie group. Conversely, in the presence of a Lie algebra of vector fields, integration gives a local Lie group action. The result now known as the third theorem provides an intrinsic and global converse to the original theorem.
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.
An invariant convex cone is a closed convex cone in the Lie algebra of a connected Lie group that ... Springer, ISBN 978-3-540-67827-4. J.-P. Serre, "Lie algebras and ...
In mathematics, specifically the theory of Lie algebras, Lie's theorem states that, [1] over an algebraically closed field of characteristic zero, if : is a finite-dimensional representation of a solvable Lie algebra, then there's a flag = = of invariant subspaces of () with =, meaning that () for each and i.
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 ...