For example, the Killing form on the Lie algebra of the Heisenberg group is identically zero, hence negative semidefinite, but this Lie algebra is not the Lie algebra of any compact group. The compact Lie algebras are classified and named according to the compact real forms of the complex semisimple Lie algebras.
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These are:. From Wikipedia, the free encyclopedia. Simple Lie groups. Other Lie groups.
Lie algebras. Exponential map Adjoint representation group algebra. Killing form Index. Semisimple Lie algebra. Dynkin diagrams Cartan subalgebra Root system Weyl group.
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Not all Lie groups are matrix groups. Consider the metaplectic group.
From wikipedia:. Therefore, the question of its explicit realization is nontrivial.
It has faithful irreducible infinite-dimensional representations, such as the Weil representation described below. However it is true that all compact Lie groups are matrix groups, as a consequence of the Peter-Weyl theorem.
This is Ado's theorem. In some sense, the Lie algebra of a Lie group captures "most" of the information about the Lie group. Finite-dimensional Lie algebras are in bijective correspondence with finite-dimensional simply-connected Lie groups.
Two Lie groups are said to be isomorphic if they are isomorphic as sets, groups, topological spaces and smooth manifolds. But conveniently enough, it is actually enough to check that they are isomorphic as topological groups, as one can show that every continuous group homomorphism between Lie groups is automatically smooth.
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I have also heard something saying that all Lie groups are in fact isomorphic to a matrix Lie group. This is not true! Another classical counterexample discussed in a paper by G. However, it is not isomorphic as Lie groups to a matrix Lie group.
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This is done in Section 4. So, no, it's not true that every Lie group is isomorphic to a matrix Lie group. Interestingly, though, it actually is true for compact Lie groups:. This is typically proved as a corollary of the famous Peter-Weyl theorem see for instance Section IV.