from its structure equations.

from a matrix algebra.

from a Lie algebra of vector fields.

from a subalgebra of an existing Lie algebra.

from a DifferentialGeometry library.

Initialize a Lie algebra and specify the labels for a basis of the Lie algebra.

View the multiplication table for a Lie algebra.

with(DifferentialGeometry): with(LieAlgebras):

StructureEquations := [[x1, x2] = x3, [x3, x1] = - x2, [x2, x3] = x1];

$\mathrm{StructureEquations}≔\left[\left[\mathrm{x1}\,\mathrm{x2}\right]\=\mathrm{x3}\,\left[\mathrm{x3}\,\mathrm{x1}\right]\=-\mathrm{x2}\,\left[\mathrm{x2}\,\mathrm{x3}\right]\=\mathrm{x1}\right]$

L := LieAlgebraData(StructureEquations, [x1, x2, x3], Alg1);

$L≔\left[\left[\mathrm{e1}\,\mathrm{e2}\right]\=\mathrm{e3}\,\left[\mathrm{e1}\,\mathrm{e3}\right]\=\mathrm{e2}\,\left[\mathrm{e2}\,\mathrm{e3}\right]\=\mathrm{e1}\right]$

lprint(L);

_DG([["LieAlgebra", Alg1, [3]], [[[1, 2, 3], 1], [[1, 3, 2], 1], [[2, 3, 1], 1]]])

DGsetup(L);

$\mathrm{Lie\; algebra:\; Alg1}$

MultiplicationTable("LieTable");

MultiplicationTable("ExteriorDerivative");

$d\left(\mathrm{\θ1}\right)\=-\mathrm{\θ2}\mathrm{`^`}\mathrm{\θ3}$

$d\left(\mathrm{\θ2}\right)\=-\mathrm{\θ1}\mathrm{`^`}\mathrm{\θ3}$

$d\left(\mathrm{\θ3}\right)\=-\mathrm{\θ1}\mathrm{`^`}\mathrm{\θ2}$

with(DifferentialGeometry): with(LieAlgebras):

M := [Matrix([[1, 0], [0, -1]]), Matrix([[0, 1], [0, 0]]), Matrix([[0, 0], [1, 0]])];

L := LieAlgebraData(M, Alg2);

$\left[\left[\mathrm{e1}\,\mathrm{e2}\right]\=2\mathrm{e2}\,\left[\mathrm{e1}\,\mathrm{e3}\right]\=-2\mathrm{e3}\,\left[\mathrm{e2}\,\mathrm{e3}\right]\=\mathrm{e1}\right]$

with(DifferentialGeometry): with(LieAlgebras):

DGsetup([x, y], N);

$\mathrm{frame\; name:\; N}$

Gamma := evalDG([D_x, D_y, x*D_x, y*D_x, x*D_y, y*D_y]);

$\mathrm{\Γ}≔\left[\mathrm{D\_x}\,\mathrm{D\_y}\,x\mathrm{D\_x}\,y\mathrm{D\_x}\,x\mathrm{D\_y}\,y\mathrm{D\_y}\right]$

L := LieAlgebraData(Gamma, Alg3);

$\left[\left[\mathrm{e1}\,\mathrm{e3}\right]\=\mathrm{e1}\,\left[\mathrm{e1}\,\mathrm{e5}\right]\=\mathrm{e2}\,\left[\mathrm{e2}\,\mathrm{e4}\right]\=\mathrm{e1}\,\left[\mathrm{e2}\,\mathrm{e6}\right]\=\mathrm{e2}\,\left[\mathrm{e3}\,\mathrm{e4}\right]\=-\mathrm{e4}\,\left[\mathrm{e3}\,\mathrm{e5}\right]\=\mathrm{e5}\,\left[\mathrm{e4}\,\mathrm{e5}\right]\=-\mathrm{e3}\+\mathrm{e6}\,\left[\mathrm{e4}\,\mathrm{e6}\right]\=-\mathrm{e4}\,\left[\mathrm{e5}\,\mathrm{e6}\right]\=\mathrm{e5}\right]$

with(DifferentialGeometry): with(LieAlgebras):

L1 := _DG([["LieAlgebra", Alg1, [4]], [[[2, 3, 1], 1], [[1, 4, 1], 2], [[2, 4, 2], 1], [[3, 4, 2], 1], [[3, 4, 3], 1]]]);

$\mathrm{L1}≔\left[\left[\mathrm{e1}\,\mathrm{e4}\right]\=2\mathrm{e1}\,\left[\mathrm{e2}\,\mathrm{e3}\right]\=\mathrm{e1}\,\left[\mathrm{e2}\,\mathrm{e4}\right]\=\mathrm{e2}\,\left[\mathrm{e3}\,\mathrm{e4}\right]\=\mathrm{e2}\+\mathrm{e3}\right]$

DGsetup(L1);

$\mathrm{Lie\; algebra:\; Alg1}$

B := evalDG([e1 + e3, e4, e2 + e4, e3]);

$B≔\left[\mathrm{e1}\+\mathrm{e3}\,\mathrm{e4}\,\mathrm{e2}\+\mathrm{e4}\,\mathrm{e3}\right]$

L2, M := LieAlgebraData(B, Alg2, "Matrix");

DGsetup(L2);

$\mathrm{Lie\; algebra:\; Alg2}$

Query(Alg2, Alg1, M, "Homomorphism");

$\mathrm{true}$

with(DifferentialGeometry): with(LieAlgebras):

L1 := _DG([["LieAlgebra", Alg1, [4]], [[[2, 3, 1], 1], [[1, 4, 1], 2], [[2, 4, 2], 1], [[3, 4, 2], 1], [[3, 4, 3], 1]]]);

$\mathrm{L1}≔\left[\left[\mathrm{e1}\,\mathrm{e4}\right]\=2\mathrm{e1}\,\left[\mathrm{e2}\,\mathrm{e3}\right]\=\mathrm{e1}\,\left[\mathrm{e2}\,\mathrm{e4}\right]\=\mathrm{e2}\,\left[\mathrm{e3}\,\mathrm{e4}\right]\=\mathrm{e2}\+\mathrm{e3}\right]$

DGsetup(L1);

$\mathrm{Lie\; algebra:\; Alg1}$

S := [e1, e2, e3];

$S≔\left[\mathrm{e1}\,\mathrm{e2}\,\mathrm{e3}\right]$

Query(S, "Subalgebra");

$\mathrm{true}$

L4 := LieAlgebraData(S, Alg4);

$\mathrm{L4}≔\left[\left[\mathrm{e2}\,\mathrm{e3}\right]\=\mathrm{e1}\right]$

with(DifferentialGeometry): with(LieAlgebras): with(Library):

References(verbose);

Gong, 1

         Classification of Nilpotent Lie Algebras of Dimension 7( Over Algebraically Closed Fields and R)

         PhD. Thesis,  University of Waterloo (1998)

Gonzalez-Lopez, 1

         Lie algebras of vector fields in the real plane (with Kamran and Olver)

         Proc. London Math Soc. Vol 64 (1992), 339--368

Kamke, 1

          Differentialgleichungen

          Chelsa Publ. Co. (1947)

Mubarakzyanov, 1

         Lie algebras of dimmensions 3, 4

         Izv. Vyssh. Uchebn. Zaved. Math 34(1963) 99.

Mubarakzyanov, 2

         Lie algebras of dimension 5

         Izv. Vyssh. Uchebn. Zaved. Math 34(1963) 99.

Mubarakzyanov, 3

         Lie algebras of dimension 6

         Izv. Vyssh. Uchebn. Zaved. Math 35(1963) 104.

Olver, 1:

         Equivalence, Invariants and Symmetry, 472--473

Petrov, 1:

         Einstein Spaces

Turkowski, 1:

         Low dimensional real Lie algebras

         JMP(29), 1990, 2139--2144

Turkowski, 2

         Solvable Lie Algebras of dimension six

         JMP(31), 1990, 1344--1350

Winternitz, 1:

         Invariants of real low dimensional Lie algebras, (with Patera, Sharp and Zassenhaus)

         JMP vol 17, No 6, June 1976, 966--994

$\left[\left[''Gong''\,1\right]\,\left[''Gonzalez-Lopez''\,1\right]\,\left[''Kamke''\,1\right]\,\left[''Morozov''\,1\right]\,\left[''Mubarakyzanov''\,1\right]\,\left[''Mubarakyzanov''\,2\right]\,\left[''Mubarakyzanov''\,3\right]\,\left[''Olver''\,1\right]\,\left[''Petrov''\,1\right]\,\left[''Turkowski''\,1\right]\,\left[''Turkowski''\,2\right]\,\left[''USU''\,2\right]\,\left[''USU''\,''2D''\right]\,\left[''Winternitz''\,1\right]\right]$

L := Browse("Winternitz", 1);

$L≔\left[\left[3\,0\right]\,\left[3\,1\right]\,\left[3\,2\right]\,\left[3\,3\right]\,\left[3\,4\right]\,\left[3\,5\right]\,\left[3\,6\right]\,\left[3\,7\right]\,\left[3\,8\right]\,\left[3\,9\right]\,\left[4\,0\right]\,\left[4\,1\right]\,\left[4\,2\right]\,\left[4\,3\right]\,\left[4\,4\right]\,\left[4\,5\right]\,\left[4\,6\right]\,\left[4\,7\right]\,\left[4\,8\right]\,\left[4\,9\right]\,\left[4\,10\right]\,\left[4\,11\right]\,\left[4\,12\right]\,\left[5\,0\right]\,\left[5\,1\right]\,\left[5\,2\right]\,\left[5\,3\right]\,\left[5\,4\right]\,\left[5\,5\right]\,\left[5\,6\right]\,\left[5\,7\right]\,\left[5\,8\right]\,\left[5\,9\right]\,\left[5\,10\right]\,\left[5\,11\right]\,\left[5\,12\right]\,\left[5\,13\right]\,\left[5\,14\right]\,\left[5\,15\right]\,\left[5\,16\right]\,\left[5\,17\right]\,\left[5\,18\right]\,\left[5\,19\right]\,\left[5\,20\right]\,\left[5\,21\right]\,\left[5\,22\right]\,\left[5\,23\right]\,\left[5\,24\right]\,\left[5\,25\right]\,\left[5\,26\right]\,\left[5\,27\right]\,\left[5\,28\right]\,\left[5\,29\right]\,\left[5\,30\right]\,\left[5\,31\right]\,\left[5\,32\right]\,\left[5\,33\right]\,\left[5\,34\right]\,\left[5\,35\right]\,\left[5\,36\right]\,\left[5\,37\right]\,\left[5\,38\right]\,\left[5\,39\right]\,\left[5\,40\right]\,\left[6\,1\right]\,\left[6\,2\right]\,\left[6\,3\right]\,\left[6\,4\right]\,\left[6\,5\right]\,\left[6\,6\right]\,\left[6\,7\right]\,\left[6\,8\right]\,\left[6\,9\right]\,\left[6\,10\right]\,\left[6\,11\right]\,\left[6\,12\right]\,\left[6\,13\right]\,\left[6\,14\right]\,\left[6\,15\right]\,\left[6\,16\right]\,\left[6\,17\right]\,\left[6\,18\right]\,\left[6\,19\right]\,\left[6\,20\right]\,\left[6\,21\right]\,\left[6\,22\right]\right]$

Browse("Winternitz", 1, L[11 .. 15]);

$''Winternitz''\,1\,\left[4\,0\right]$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$''Winternitz''\,1\,\left[4\,1\right]$

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

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$''Winternitz''\,1\,\left[4\,2\right]$

$\left[\left[\mathrm{e1}\,\mathrm{e4}\right]\=a\mathrm{e1}\,\left[\mathrm{e2}\,\mathrm{e4}\right]\=\mathrm{e2}\,\left[\mathrm{e3}\,\mathrm{e4}\right]\=\mathrm{e2}\+\mathrm{e3}\right]$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$''Winternitz''\,1\,\left[4\,3\right]$

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

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$''Winternitz''\,1\,\left[4\,4\right]$

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

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

L := Retrieve("Winternitz", 1, [4, 4], Alg1);

$L≔\left[\left[\mathrm{e1}\,\mathrm{e4}\right]\=\mathrm{e1}\,\left[\mathrm{e2}\,\mathrm{e4}\right]\=\mathrm{e1}\+\mathrm{e2}\,\left[\mathrm{e3}\,\mathrm{e4}\right]\=\mathrm{e2}\+\mathrm{e3}\right]$

DGsetup(L);

$\mathrm{Lie\; algebra:\; Alg1}$

with(DifferentialGeometry): with(LieAlgebras):

M := [Matrix([[1, 0], [0, -1]]), Matrix([[0, 1], [0, 0]]), Matrix([[0, 0], [1, 0]])];

L := LieAlgebraData(M, Alg1);

$L≔\left[\left[\mathrm{e1}\,\mathrm{e2}\right]\=2\mathrm{e2}\,\left[\mathrm{e1}\,\mathrm{e3}\right]\=-2\mathrm{e3}\,\left[\mathrm{e2}\,\mathrm{e3}\right]\=\mathrm{e1}\right]$

DGsetup(L, [E], [epsilon]);

$\mathrm{Lie\; algebra:\; Alg1}$

MultiplicationTable("LieTable");

MultiplicationTable("ExteriorDerivative");

$d\left(\mathrm{\ε1}\right)\=-\mathrm{\ε2}\mathrm{`^`}\mathrm{\ε3}$

$d\left(\mathrm{\ε2}\right)\=-2\mathrm{\ε1}\mathrm{`^`}\mathrm{\ε2}$

$d\left(\mathrm{\ε3}\right)\=2\mathrm{\ε1}\mathrm{`^`}\mathrm{\ε3}$

DGsetup(L, [X, Y, Z], [alpha, beta, sigma]);

$\mathrm{Lie\; algebra:\; Alg1}$

MultiplicationTable("LieTable");

MultiplicationTable("ExteriorDerivative");

$d\left(\mathrm{\α}\right)\=-\mathrm{\β}\mathrm{`^`}\mathrm{\σ}$

$d\left(\mathrm{\β}\right)\=-2\mathrm{\α}\mathrm{`^`}\mathrm{\β}$

$d\left(\mathrm{\σ}\right)\=2\mathrm{\α}\mathrm{`^`}\mathrm{\σ}$