LieAlgebras[ChangeGradedComponent] - change one or more components of a graded Lie algebra
Calling Sequences
ChangeGradedComponent(alg, newcomponent, newalg)
Parameters
alg - a name or string, the name of an initialized Lie algebra 𝔤
newcomponent - a list, specifying the new graded components
newalg - a name or string, the name of a new graded Lie algebra to be created
Description
See Also
Let 𝔤 be a graded Lie algebra with (for example) grading 𝔤 = 𝔤−2 ⊕ 𝔤−1 ⊕ 𝔤0 ⊕ 𝔤1⊕ 𝔤2⊕ 𝔤3 . With newcomponent given by (for example) [2 = h], where h is a list of vectors in 𝔤2, the command ChangeGradedComponent will return the structure equations for the new graded Lie algebra𝔤 = 𝔤−2 ⊕ 𝔤−1⊕ 𝔤0 ⊕ 𝔤1 ⊕ h⊕ 𝔤3 .
Examples
with(DifferentialGeometry): with(LieAlgebras):
Example 1.
Define a 9-dimensional Lie algebra alg1 with grading 𝔤−3 ⊕ 𝔤−2⊕ 𝔤−1⊕ 𝔤0, where 𝔤−3= e1, e2, 𝔤−2 = e3, 𝔤−1 = e4 ,e5 and 𝔤0 = e6 ,e7 , e8, e9. Here are the structure equations:
StrEq := [[x1, x6] = -x1, [x1, x8] = -x2, [x2, x7] = -x1, [x2, x9] = -x2, [x3, x4] = -x1, [x3, x5] = -x2, [x3, x6] = -(1/3)*x3, [x3, x9] = -(1/3)*x3, [x4, x5] = x3, [x4, x6] = -(2/3)*x4, [x4, x8] = -x5, [x4, x9] = (1/3)*x4, [x5, x6] = (1/3)*x5, [x5, x7] = -x4, [x5, x9] = -(2/3)*x5, [x6, x7] = x7, [x6, x8] = -x8, [x7, x8] = x6-x9, [x7, x9] = x7, [x8, x9] = -x8];
StrEq:=x1,x6=−x1,x1,x8=−x2,x2,x7=−x1,x2,x9=−x2,x3,x4=−x1,x3,x5=−x2,x3,x6=−13x3,x3,x9=−13x3,x4,x5=x3,x4,x6=−23x4,x4,x8=−x5,x4,x9=13x4,x5,x6=13x5,x5,x7=−x4,x5,x9=−23x5,x6,x7=x7,x6,x8=−x8,x7,x8=x6−x9,x7,x9=x7,x8,x9=−x8
Use the keyword grading to specify the grading of this algebra. Initialize.
LD1 := LieAlgebraData(StrEq, [x1, x2, x3, x4, x5, x6, x7, x8, x9], alg1, grading = [-3, -3, -2, -1, -1, 0, 0, 0, 0]);
LD1:=e1,e6=−e1,e1,e8=−e2,e2,e7=−e1,e2,e9=−e2,e3,e4=−e1,e3,e5=−e2,e3,e6=−13e3,e3,e9=−13e3,e4,e5=e3,e4,e6=−23e4,e4,e8=−e5,e4,e9=13e4,e5,e6=13e5,e5,e7=−e4,e5,e9=−23e5,e6,e7=e7,e6,e8=−e8,e7,e8=e6−e9,e7,e9=e7,e8,e9=−e8
DGsetup(LD1);
Lie algebra: alg1
Note that the vectors e6, e7 define a 2-dimensional subalgebra of 𝔤0.
LieBracket(e6, e7);
e7
Therefore we can replace all of 𝔤0 with just e6, e7. The result is a 7-dimensional graded Lie algebra which is a sub-algebra of the one we started with.
newLD1a := ChangeGradedComponent(alg1, [0 = [e6, e7]], newalg1);
newLD1a:=e1,e6=−e1,e2,e7=−e1,e3,e4=−e1,e3,e5=−e2,e3,e6=−13e3,e4,e5=e3,e4,e6=−23e4,e5,e6=13e5,e5,e7=−e4,e6,e7=e7
DGsetup(newLD1a);
Lie algebra: newalg1
Tools:-DGinfo("Grading");
−3,−3,−2,−1,−1,0,0
DifferentialGeometry, LieAlgebras, TanakaProlongation
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