Proving Linearity of Simple Lie Groups with Trivial Center

In summary, a simple Lie Group is a connected and simply connected mathematical group without any non-trivial normal subgroups. It is considered linear if it can be represented as a subgroup of the general linear group GL(n) of invertible matrices. Some key properties of a simple Lie Group include being connected, compact, and simply connected, with a finite center and non-abelian group operation. Examples include SU(n), SO(n), Sp(n), E8, and G2. Simple Lie Groups are important in mathematics as a fundamental building block and have various applications in areas such as symmetry and differential equations.
  • #1
shybishie
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Suppose we have a simple Lie Group [itex]G[/itex], i.e, a Lie Group with a trivial center(the identity). Show that this group must be linear, i.e, we can map it to a Lie subgroup of [itex]GL(N)[/itex].So far, I have that from abstract algebra we can show a group with trivial center is isomorphic to the inner automorphisms on the group, let us say [itex]Inn(G)[/itex]. Also, [itex]Inn(G)[/itex] is clearly a group of diffeomorphisms on [itex]G[/itex]. I have a vague intuition that if we can show [itex]Inn(G)[/itex] is linear , that would show our desired result, although I am not sure how to make this more concrete.

Does anyone have a few hints that might get me on the right track, or tools that I am missing? I am pondering if I can tie this to the Lie algebra of [itex]G[/itex] in some manner.

Also to clarify, this is not part of a graded problem set. I saw it on an assignment some months ago, but didn't figure it out at the time and I want to think my way through it now. The forum rules say that graduate level coursework questions can be posted here, but if it's a problem I apologize for the misclassification.
 
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Answer:One way to show that a simple Lie group G must be linear is by using the fact that every Lie group can be viewed as a differentiable manifold, and thus its tangent space at each point can be identified with the Lie algebra of the group. Since G is a simple Lie group, it has a trivial center, which implies that its Lie algebra is semisimple, i.e., it is the direct sum of simple Lie algebras. By the Ado-Iwasawa Theorem, any semisimple Lie algebra can be mapped to a linear Lie subalgebra of GL(N). This mapping can then be extended to a map from G to a linear Lie subgroup of GL(N), since the exponential map is a diffeomorphism between the Lie algebra and the Lie group.
 

1. What is a simple Lie Group?

A simple Lie Group is a type of mathematical group that is both connected and simply connected. This means that it is a continuous group that does not have any non-trivial normal subgroups.

2. What does it mean for a simple Lie Group to be linear?

A simple Lie Group is considered linear if it can be represented as a subgroup of the general linear group GL(n) of invertible matrices of size n. This allows for the use of linear algebra techniques in the study of the group.

3. What are the properties of a simple Lie Group?

Some key properties of a simple Lie Group include being connected, compact, and simply connected. It also has a finite center and is non-abelian, meaning that its group operation is not commutative.

4. What are some examples of simple Lie Groups?

Some examples of simple Lie Groups include SU(n), SO(n), and Sp(n), which are special linear groups of unitary, orthogonal, and symplectic matrices, respectively. Other examples include the exceptional Lie groups such as E8 and G2.

5. Why are simple Lie Groups important in mathematics?

Simple Lie Groups are important in mathematics because they serve as a fundamental building block for more complex groups. They also have many applications in various areas of mathematics, physics, and engineering, such as in the study of symmetry and differential equations.

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