Ideals, Simple and Semisimple Lie Algebras

Lie Algebra Ideals

Lie Algebra Ideal and Quotients

A subspace of a Lie algebra is an ideal, denoted , if . That is, for any and , we have .
If , then the quotient is well-defined, and .

e.g.

1. For any homomorphism , the kernel is an ideal of .
2. The center of , , is an ideal.
3. The centralizer of a subset , , is an ideal if is an ideal. (This is sufficient but not necessary)
4. The commutator algebra of , , is an ideal.
5. For ideals , their sum , and their Lie bracket are also ideals.
   

Simple Lie Algebras

A Lie algebra is simple if it has no non-trivial ideals.

e.g. The Lie algebra with brackets , , is simple provided .
Proof Let be a non-zero ideal. By the commutation relation, we know that the adjoint map is diagonalizable with eigenvalues corresponding to eigenvectors . Take any non-zero element .
If , apply twice: and . Since and , this implies .
Similarly, if , applying twice gives , showing .
Once or is in , we have . Then and show that if one of is in , the other must be as well.
If , then with , so , yielding a contradiction.

Isomorphism Theorems for Lie Algebras

The following isomorphism theorems hold for Lie algebras, similar to those for groups and rings:

• Suppose is a Lie algebra homomorphism, then .
• If and are ideals, then .
• If , then .
  

Proposition

If is an ideal, then , and .

Proof Suppose and , let , then So , hence . Now consider the quotient map . We have and . Thus, by the first isomorphism theorem, .

Solvability and Semi-Simplicity

Derived Series and Solvability

The derived series of a Lie algebra is a sequence of ideals defined recursively as follows:

A Lie algebra is solvable if for some integer .

e.g. The Borel subalgebra of , consisting of all upper triangular matrices, is solvable.

Semi-Simple Lie Algebras

A Lie algebra is semi-simple if its only solvable ideal is .

Properties of Solvable Lie Algebras

Let be a Lie algebra.

1. If is solvable, so are its subalgebras and homomorphic images.
2. If is a solvable ideal and the quotient is solvable, then is solvable.
3. If are solvable ideals, then their sum is also a solvable ideal.
   

Proof

1. Suppose is a subalgebra. Then for all . If is solvable, then for some , so , hence is solvable. If is a homomorphism, then for all . If is solvable, then for some , so , hence is solvable.
2. Consider the quotient map . We have for all . If and are solvable, then and for some . From the proposition, we know that , so , is solvable.
3. We note that , so by part 2, is solvable.

This implies there exists a unique maximal solvable ideal in :

Radical of a Lie Algebra

For any Lie algebra , there exists a unique maximal solvable ideal in , called the radical of , denoted . The quotient algebra is semi-simple.

Nilpotent Lie Algebras

Lower Central Series and Nilpotency

The lower central series of a Lie algebra is defined recursively:

A Lie algebra is nilpotent if for some integer .

e.g.

• The Lie algebra of strictly upper triangular matrices is nilpotent. However, the Borel subalgebra of all upper triangular matrices is solvable but not nilpotent.
• The affine Lie algebra with is solvable but not nilpotent.
  

Proposition

Nilpotency implies solvability.

Proof: for all .

Properties of Nilpotent Lie Algebras

Let be a Lie algebra.

1. Subalgebras and homomorphic images of a nilpotent Lie algebra are nilpotent.
2. If is nilpotent, then is nilpotent.
3. If is nilpotent and non-zero, then its center is non-zero.
   

Proof (1) is clear, we will show (2) and (3).
(2) Let be the quotient map. Suppose . Then . So . Thus, . Hence, is nilpotent.
(3) Suppose is nilpotent, non-zero and . Assume that . This means , so . Repeating this argument, we get , a contradiction. Hence, .