where  and *.  

If  and  are the components of and  with respect to a basis for (and dual basis for ), then  .

Example 1.

We construct the Lie algebras  and  (3)4(2)  as subalgebras of (5). First, here are the 5×5 skew-symmetric matrices which define

Calculate the structure equations and initialize.

Define the representation space .  We shall define the invariant tensors we need on

The standard inclusion of in is given as the subalgebra of matrices which fix the vectors and   .

Comparing with the matrices in A, we see this is precisely the subalgebra we want.

We can define in as the subalgebra which preserves the subspaces spanned by] and .  

Example 2.

The computation of Example 1 can be done with the other calling sequences.

1. With a representation.

2. With a Lie algebra of vector fields.

3. With a Lie algebra constructed using the procedure SimpleLieAlgebraData .

Example 3.

Calculate the subalgebra of consisting of 2×2 block upper triangular matrices. First initialize the Lie algebra of all matrices.

Define the representation space.

The matrices we want preserve the following subspaces of

We can see what matrices these correspond to in several ways.  One method is to first form a general linear combination of the vectors in .

Now calculate the matrix associated to   in the standard representation.

Example 4.

In this example we calculate the intersection These are the skew-symmetric matrices which also preserve a non-degenerate 2-form. Then we show that this intersection is isomorphic to  First we initialize the Lie algebra for

Now define an 8-dimensional representation space and a 2-form on

Find the subalgebra of  which preserves this 2-form.

Here are the explicit matrices.

Check that the matrices belong to

Check that the matrices belong to .

The isomorphism to  is given by   =  where  are 4×4 matrices.  We use the command SubMatrix  to construct this map.

Check that each of these matrices belong to .

Finally, we see that the structure equations for these two matrix algebras are identical.

Example 5.

The compact real form of the exceptional Lie algebra as the subalgebra of can be computed using the command MatrixAlgebras. First we initialize the Lie algebra  

Now define a7-dimensional representation space and a 3-form on

Calculate the subalgebra of   which leaves the 3-form invariant.

Here are the explicit matrices.

The Lie algebra defined by either the vectors or the matrices   is a 14-dimensional Lie algebra with negative-definite Killing form and 2-dimensional Cartan subalgebra.

Example 6.

The split real form of the exceptional Lie algebra as the subalgebra of (4, 3)  is similarly computed.

Now define a 7-dimensional representation space and a 3-form on

Calculate the subalgebra of  which  leaves the 3-form invariant.

Here are the explicit matrices.

The Lie algebra defined by either the vectors or the matrices   is a 14-dimensional Lie algebra.

DifferentialGeometry, CartanSubalgebra, Killing, Query[MatrixAlgebra], Representation, SimpleLieAlgebraData, StandardRepresentation