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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

Description

• 

Let 𝔤 be a graded Lie algebra with (for example) grading 𝔤 = 𝔤2  𝔤1  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 = 𝔤2  𝔤1 𝔤0  𝔤1  h 𝔤3 .

Examples

with(DifferentialGeometry): with(LieAlgebras):

 

Example 1.

Define a 9-dimensional Lie algebra alg1 with grading 𝔤3  𝔤2 𝔤1 𝔤0, where 𝔤3= e1, e2, 𝔤2 = e3, 𝔤1 = 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=x6x9,x7,x9=x7,x8,x9=x8

(1)

 

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=e6e9,e7,e9=e7,e8,e9=e8

(2)

DGsetup(LD1);

Lie algebra: alg1

(3)

 

Note that the vectors e6, e7 define a 2-dimensional subalgebra of 𝔤0.

alg1 > 

LieBracket(e6, e7);

e7

(4)

 

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.

alg1 > 

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

(5)
alg1 > 

DGsetup(newLD1a);

Lie algebra: newalg1

(6)
alg1 > 

Tools:-DGinfo("Grading");

3,3,2,1,1,0,0

(7)

See Also

DifferentialGeometry, LieAlgebras, TanakaProlongation

 


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