A semi-direct sum for a Lie algebra $\mathrm{\𝔤}$ is a vector space decomposition $\mathrm{\𝔤} \= h \+ k \,$where $h \subset \mathrm{\𝔤}$ is an ideal and $k\mathit{ }\mathit{\subset }\mathrm{\𝔤}\mathit{ }$is a subalgebra. For each $y \in k$, the adjoint matrix ad$\left(y\right)$ restricts to a derivation on $h\.$Conversely, given Lie algebras $h$ and $k$and a Lie algebra homomorphism $\mathrm{\φ} \:k \to$der($h\)$, a Lie bracket can be defined on the vector space $\mathrm{\𝔤}\mathit{ }\mathit{\=}\mathit{ }h \+k$by setting

To create a semi-direct sum of $h$and $k$using the program SemiDirectSum, first use the program Derivations to calculate the matrix algebra of derivations for $h$. Then use the program LieAlgebraData to obtain the Lie algebra data structure for the algebra of derivations der($h\)$. Then use the transformation program to create a homomorphism from $k$to der($h\)$. This gives all the data needed for the SemiDirectSum program.

The program SemiDirectSum returns the Lie algebra data structure for the semi-direct sum. A Lie algebra data structure contains the structure constants in a standard format used by the LieAlgebras package. In the LieAlgebras package, the command DGsetup is used to initialize a Lie algebra -- that is, to define the basis elements for the Lie algebra and its dual and to store the structure constants for the Lie algebra in memory.

The command SemiDirectSum is part of the DifferentialGeometry:-LieAlgebras package. It can be used in the form SemiDirectSum(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-SemiDirectSum(...).

with(DifferentialGeometry): with(LieAlgebras):

L1 := _DG([["LieAlgebra", Alg1, [3]], [[[1, 3, 1], 1], [[2, 3, 1], 1], [[2, 3, 2], 1]]]);

$\mathrm{L1}:=\left[\left[\mathrm{e1}\,\mathrm{e3}\right]\=\mathrm{e1}\,\left[\mathrm{e2}\,\mathrm{e3}\right]\=\mathrm{e1}\+\mathrm{e2}\right]$

DGsetup(L1, [x], [a]):

L2 := _DG([["LieAlgebra", Alg2, [2]],[[[1, 2, 1], 1]]]):

DGsetup(L2, [y], [b]):

MultiplicationTable(Alg1, "LieTable"), MultiplicationTable(Alg2, "LieTable");

ChangeLieAlgebraTo(Alg1):

Derivations("Full");

Der := map(Matrix, [[[1, 0, 0], [0, 1, 0], [0, 0, 0]], [[0, 1, 0], [0, 0, 0], [0, 0, 0]], [[0, 0, 1], [0, 0, 0], [0, 0, 0]], [[0, 0, 0], [0, 0, 1], [0, 0, 0]]]);

L3 := LieAlgebraData(Der, Alg3);

$\mathrm{L3}:=\left[\left[\mathrm{e1}\,\mathrm{e3}\right]\=\mathrm{e3}\,\left[\mathrm{e1}\,\mathrm{e4}\right]\=\mathrm{e4}\,\left[\mathrm{e2}\,\mathrm{e4}\right]\=\mathrm{e3}\right]$

DGsetup(L3, [E], [c]):

MultiplicationTable("LieTable");

Phi := Transformation([[y1, E4], [y2, (-1) &mult E1]]);

$\mathrm{\Φ}:=\left[\left[\mathrm{y1}\,\mathrm{E4}\right]\,\left[\mathrm{y2}\,-\mathrm{E1}\right]\right]$

Query(Alg2, Alg3, Phi, "Homomorphism");

$\mathrm{true}$

L4 := SemiDirectSum(Alg1, Alg2, Phi, Der, Alg4);

$\mathrm{L4}:=\left[\left[\mathrm{e1}\,\mathrm{e3}\right]\=\mathrm{e1}\,\left[\mathrm{e1}\,\mathrm{e5}\right]\=\mathrm{e1}\,\left[\mathrm{e2}\,\mathrm{e3}\right]\=\mathrm{e1}\+\mathrm{e2}\,\left[\mathrm{e2}\,\mathrm{e5}\right]\=\mathrm{e2}\,\left[\mathrm{e3}\,\mathrm{e4}\right]\=-\mathrm{e2}\,\left[\mathrm{e4}\,\mathrm{e5}\right]\=\mathrm{e4}\right]$

DGsetup(L4);

$\mathrm{Lie\; algebra:\; Alg4}$

MultiplicationTable("LieTable");

Query(Alg4, "Jacobi");

$\mathrm{true}$

Query([e1, e2, e3], "Ideal"), Query([e4, e5], "Subalgebra");

$\mathrm{true}\,\mathrm{true}$

ApplyHomomorphism(Phi, y1), Der[4], Adjoint(e4, [e1, e2, e3]);

ApplyHomomorphism(Phi, y2), Der[1], Adjoint(e5, [e1, e2, e3]);