Get On Complex Lie Algebras With A Simple Real Form - Heldermann-Verlag - Heldermann-verlag

A Lie group is first of all a group. Secondly it is a smooth manifold which is a specific kind of geometric object. The circle and the sphere are examples of smooth manifolds. Finally the algebraic structure and the geometric structure must be compatible in a precise way.

It is a basic fact in the structure theory of complex semisimple Lie algebras that every such algebra has two special real forms: one is the compact real form and corresponds to a compact Lie group under the Lie correspondence (its Satake diagram has all vertices blackened), and the other is the split real form and ...

In mathematics, the complexification or universal complexification of a real Lie group is given by a continuous homomorphism of the group into a complex Lie group with the universal property that every continuous homomorphism of the original group into another complex Lie group extends compatibly to a complex analytic ...

It is a basic fact in the structure theory of complex semisimple Lie algebras that every such algebra has two special real forms: one is the compact real form and corresponds to a compact Lie group under the Lie correspondence (its Satake diagram has all vertices blackened), and the other is the split real form and ... Real form (Lie theory) - Wikipedia wikipedia.org https://en.wikipedia.org › wiki › Real_form_(Lie_theory) wikipedia.org https://en.wikipedia.org › wiki › Real_form_(Lie_theory)

Lie groups are classified ing to their algebraic properties (simple, semisimple, solvable, nilpotent, abelian), their connectedness (connected or simply connected) and their compactness.

A complex Lie group is a Lie group that is a group object not just internal to smooth manifolds but in fact to complex manifolds. Hence it is a complex manifold G equipped with a group structure such that both the multiplication map G × G → G G \times G \to G as well as the inverse map G → G are holomorphic functions. complex Lie group in nLab ncatlab.org https://ncatlab.org › nlab › show › complex+Lie+group ncatlab.org https://ncatlab.org › nlab › show › complex+Lie+group