LieAlgebras[Derivations] - find the inner and/or Outer derivations of a Lie algebra
Calling Sequences
Derivations(Algname, "keyword")
Parameters
Algname - (optional) name or string, the name of a Lie algebra
keyword - one of the 3 keywords "Inner", "Full", or "Outer"
|
Description
|
|
•
|
A matrix A is a derivation for g if the associated linear transformation mapping g to g satisfies A([x, y]) = [A(x), y] + [x, A(y)] for all x, y in g. The set of all derivations defines a matrix Lie algebra Derivations(g). For each x in g the matrix A= ad(x) defines a derivation -- these are the inner derivations Derivations(g, "Inner"). The inner derivations define an ideal in Derivations(g) and the quotient Lie algebra Derivations(g)/Derivations(g, "Inner") is the Lie algebra of outer derivations.
|
•
|
Derivations(Algname, "Inner") returns a list of linearly independent matrices which defines a basis for the Lie algebra of inner derivations for the Lie algebra Algname.
|
•
|
Derivations(Algname, "Full") returns a list of linearly independent matrices which defines a basis for the Lie algebra of all derivations for the Lie algebra Algname.
|
•
|
Derivations(Algname, "Outer") returns a list of linearly independent matrices which gives a representative list of the outer derivations for the Lie algebra Algname.
|
•
|
If the first (optional) argument Algname is missing, then the derivations of the current Lie algebra are computed.
|
•
|
The command Derivations is part of the DifferentialGeometry:-LieAlgebras package. It can be used in the form Derivations(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-Derivations(...).
|
|
|
Examples
|
|
>
|
|
Example 1.
First initialize a Lie algebra and display the Lie bracket multiplication table.
>
|
|
>
|
|
Alg1 >
|
|
| (2.1) |
For the Lie algebra Alg1 we find that Derivations(Alg1, "Inner") is 4 dimensional and Derivations(Alg1) is 8 dimensional.
Alg1 >
|
|
| (2.2) |
Alg1 >
|
|
| (2.3) |
Alg1 >
|
|
| (2.4) |
We can study the properties of Derivations(Alg1) by initializing these matrices as a Lie algebra. We use as a basis for Derivations(Alg1) the inner and outer derivations.
Alg1 >
|
|
Alg1 >
|
|
| (2.5) |
Alg1 >
|
|
We see that the derivation algebra is solvable.
DerAlg >
|
|
| (2.6) |
We check that the vectors [E1, E2, E3, E4] (corresponding to the inner derivations) define an ideal.
DerAlg >
|
|
| (2.7) |
We compute the quotient algebra of outer derivations.
DerAlg >
|
|
| (2.8) |
DerAlg >
|
|
| (2.9) |
|
|