Example 1.

First initialize a Lie algebra.  We illustrate the fact that Ad(x), for any x in the Lie algebra, is always a Lie algebra homomorphism (isomorphism).

Example 2.

The Matrix exponential of any Outer derivation is also a Lie algebra homomorphism (isomorphism).

Example 3.

In this example we construct the quotient algebra of Alg1 by the ideal [e1].  Call the quotient Alg2.  We check that the canonical projection map from Alg1 to Alg2 is a Lie algebra homomorphism.

The following Matrix A3 maps e1 -> 0, e2 -> x1, e3 -> x2, e4 -> x3.

Example 4.

In this example we shall find all the monomorphisms from the 2 dimensional solvable Lie algebra into Alg1.

This effectively computes all the 2 dimensional non-Abelian subalgebras of Alg1.

First initialize the 2 dimensional solvable algebra and call it Alg3.

Define a matrix A4 representing an arbitrary linear transformation from Alg1 to Alg2.

Determine the parameter values for which A4 is a Lie algebra homomorphism.

The equations that must hold for A4 to define a Lie algebra homomorphism are given by EQ.

The possible Lie algebra homomorphisms are given by B.  Note that B[2], B[3] and B[4] can be chosen to be full rank and therefore define Lie algebra isomorphisms.

DifferentialGeometry, LieAlgebras, Adjoint, Derivations, Query, QuotientAlgebra, Transformation