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LieAlgebras[SimpleLieAlgebraData] - obtain the structure equations for a classical matrix Lie algebra

Calling Sequences

     SimpleLieAlgebraData(algtype, algname, option)

Parameters

     algtype   - a string, describing the type and dimension of a classical matrix algebra

      algname   - an unassigned name or a string, the name of the classical matrix algebra to be constructed

     options   - (optional) keyword arguments labelformat, labels which specify the labelling of the basis for the Lie algebra. Different standard basis for some of the Lie algebras can be specified with the keyword version.

 

Description

Examples

Description

 This command returns the structure equations (see LieAlgebraData) for any one of the following Lie algebras:

 

 Type

Lie algebra

A

  sln, sup, q (two versions), su*n, pq

B

sop, q,  p+q =2 m+1 (two versions), p q

C

 spn, , spp, q, spn, pq

D

sop, q, p+q =2 m (two versions), so*n, n = 2 m, pq

 F

 f4, 52, 0 or f4, compact, f4, 36, 16, f4, 28, 24or f4, Split

 G

 g2, 14, 0 or g2, compact, g2, 6, 8or g2, split (two versions)

Other

 gln, ℝ, gln, ℂ, sln,, up, q, son,, spn,ℂ, soln, niln 

• 

The Lie algebras A,B, C, D, F, G are all simple Lie algebras. The Other are classical matrix algebras which are often used in Lie theory and differential geometry.

• 

The precise definitions and examples of each of these Lie algebras are found in SimpleLieAlgebraDataDetails .

• 

The command StandardRepresentation generates the standard matrix representations of these algebras.

• 

Cartan matrices, Dynkin diagram, Satake diagrams, positive roots can easily be found for each of the simple Lie algebra. See also SimpleLieAlgebraProperties .

• 

Subalgebras of any of these Lie algebras can be calculate using the command MatrixSubalgebras.

• 

 Two versions of the Lie algebras sup, q and sop, q are available, corresponding the choices

Q1  =  0Iq0Iq0000Ipq     or    Q2 =  Ip00Iq  

for the quadratic form preserved by these algebras. The keyword argument version.specifies the choice. The default is version =1. This choice is preferred for roots space computations.

• 

The keyword arguments labelformat, labels  allow for the labeling of the basis of the abstract Lie algebra which characterizes the basis elements in terms of their standard matrix elements. See LieAlgebraData, DGsetup.

Examples

with(DifferentialGeometry): with(LieAlgebras):

 

Example 1.

Initialize the Lie algebra sl3, the Lie algebra of trace-free 3×3 matrices.

LD1 := SimpleLieAlgebraData("sl(3)", alg1, labelformat = "gl", labels = ['E', 'theta']);

LD1:=e1,e3=e3,e1,e4=2e4,e1,e5=e5,e1,e6=e6,e1,e7=2e7,e1,e8=e8,e2,e3=e3,e2,e4=e4,e2,e5=e5,e2,e6=2e6,e2,e7=e7,e2,e8=2e8,e3,e5=e1e2,e3,e6=e4,e3,e7=e8,e4,e5=e6,e4,e7=e1,e4,e8=e3,e5,e8=e7,e6,e7=e5,e6,e8=e2,E11,E22,E12,E13,E21,E23,E31,E32,θ11,θ22,θ12,θ13,θ21,θ23,θ31,θ32

(2.1)

 

When this output is passed to DGsetup, the 8-dimensional Lie algebra with the foregoing structure equations is initialized and the unassigned names E11 ,E22, ... θ11, θ22, .. are assigned as vectors and 1-forms for this Lie algebra.

DGsetup(LD1);

Lie algebra: alg1

(2.2)

 

Here is the Lie bracket multiplication table for sl3.

alg1 > 

MultiplicationTable("LieTable");

| E11E22E12E13E21E23E31E32------------------------------------E11| 00E122E13E21E232E31E32E22| 00E12E13E212E23E312E32E12| E12E1200E11E22E13E320E13| 2E13E1300E230E11E12E21| E21E21E11+E22E23000E31E23| E232E23E13000E21E22E31| 2E31E31E32E110E2100E32| E322E320E12E31E2200

(2.3)

 

This coincides with the commutator formulas for the standard matrix representation of sl3.

alg1 > 

StandardRepresentation(alg1);

100000001,000010001,010000000,001000000,000100000,000001000,000000100,000000010

(2.4)

The 3rd basis element E12 matches the 3rd matrix in the standard representation and is precisely the elementary matrix with a 1 in the 1st column, 2nd row.

 

Example 2

We initialize the Lie algebra so3,1 in two different basis. Our first version is:

alg1 > 

LD2a := SimpleLieAlgebraData("so(3, 1)", so31a, labelformat = "gl", labels = ['X', 'zeta']);

LD2a:=e1,e2=e2,e1,e3=e3,e1,e4=e4,e1,e5=e5,e2,e4=e1,e2,e5=e6,e2,e6=e3,e3,e4=e6,e3,e5=e1,e3,e6=e2,e4,e6=e5,e5,e6=e4,X11,X13,X14,X23,X24,X34,ζ11,ζ13,ζ14,ζ23,ζ24,ζ34

(2.5)
alg1 > 

DGsetup(LD2a);

Lie algebra: so31a

(2.6)
so31a > 

MultiplicationTable("LieTable");

| X11X13X14X23X24X34----------------------------X11| 0X13X14X23X240X13| X1300X11X34X14X14| X1400X34X11X13X23| X23X11X3400X24X24| X24X34X1100X23X34| 0X14X13X24X230

(2.7)

 

Our second version is :

so31a > 

LD2b := SimpleLieAlgebraData("so(3, 1)", so31b, labelformat = "gl", labels = ['Y', 'xi'], version = 2);

LD2b:=e1,e2=e3,e1,e3=e2,e1,e4=e5,e1,e5=e4,e2,e3=e1,e2,e4=e6,e2,e6=e4,e3,e5=e6,e3,e6=e5,e4,e5=e1,e4,e6=e2,e5,e6=e3,Y12,Y13,Y23,Y14,Y24,Y34,ξ12,ξ13,ξ23,ξ14,ξ24,ξ34

(2.8)
alg1 > 

DGsetup(LD2b);

Lie algebra: so31b

(2.9)
so31a > 

MultiplicationTable("LieTable");

| Y12Y13Y23Y14Y24Y34----------------------------Y12| 0Y23Y13Y24Y140Y13| Y230Y12Y340Y14Y23| Y13Y1200Y34Y24Y14| Y24Y3400Y12Y13Y24| Y140Y34Y120Y23Y34| 0Y14Y24Y13Y230

(2.10)

 

From the standard matrix representations for these 2 Lie algebras we can construct a Lie algebra isomorphism Φ: so31a  so31b. First let us define the quadratic forms used in each version.

so31b > 

with(LinearAlgebra):

so31b > 

Qa := Matrix([[0, 1, 0, 0], [1, 0, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]]);

Qa:=0100100000100001

(2.11)
so31b > 

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

Qb:=1000010000100001

(2.12)

 

Here is the change of basis matrix relating Qa to Qb.

so31b > 

P := Matrix([[0,0,1/sqrt(2), 1/sqrt(2)],[0,0,1/sqrt(2), -1/sqrt(2)], [0, 1, 0, 0], [1, 0, 0, 0]]);

P:=001221220012212201001000

(2.13)
so31b > 

Transpose(P).Qa.P;

1000010000100001

(2.14)

 

Here is the standard representation for so3,1 using Qa.

so31b > 

A := StandardRepresentation(so31a);

A:=1000010000000000,0010000001000000,0001000000000100,0000001010000000,0000000100001000,0000000000010010

(2.15)

 

Here is the standard representation for so3,1 using Qb.

so31b > 

B := StandardRepresentation(so31b);

B:=0100100000000000,0010000010000000,0000001001000000,0001000000001000,0000000100000100,0000000000010010

(2.16)

 

Here are the matrices A under the change of basis defined by P. Since the resulting matrices now preserve Qb, they must be linear combinations of the matrices B.

so31b > 

A1 := [seq(P^(-1).a.P, a = A)];

A1:=0000000000010010,000000122122012200012200,001221220000122000122000,000000122122012200012200,001221220000122000122000,0100100000000000

(2.17)
so31b > 

C := GetComponents(A1, B);

C:=0,0,0,0,0,1,0,0,122,0,122,0,0,122,0,122,0,0,0,0,122,0,122,0,0,122,0,122,0,0,1,0,0,0,0,0

(2.18)

 

These components specify the matrix of the isomorphism we want.

so31b > 

Phi := Transformation(so31a, so31b, Transpose(Matrix(C)));

Φ:=X11,Y34,X13,122Y23+122Y24,X14,122Y13+122Y14,X23,122Y23122Y24,X24,122Y13122Y14,X34,Y12

(2.19)
so31b > 

Query(Phi, "Homomorphism");

true

(2.20)

 

Example 3.

Two versions of the split real form of the exception Lie algebra g2 are available. The first version gives a Chevalley basis.

LD3a := SimpleLieAlgebraData("g(2, Split)", g2a, version = 1);

LD3a:=e1,e3=2e3,e1,e4=3e4,e1,e5=e5,e1,e6=e6,e1,e7=3e7,e1,e9=2e9,e1,e10=3e10,e1,e11=e11,e1,e12=e12,e1,e13=3e13,e2,e3=e3,e2,e4=2e4,e2,e5=e5,e2,e7=e7,e2,e8=e8,e2,e9=e9,e2,e10=2e10,e2,e11=e11,e2,e13=e13,e2,e14=e14,e3,e4=e5,e3,e5=2e6,e3,e6=3e7,e3,e9=e1,e3,e11=3e10,e3,e12=2e11,e3,e13=e12,e4,e7=e8,e4,e10=e2,e4,e11=e9,e4,e14=e13,e5,e6=3e8,e5,e9=3e4,e5,e10=e3,e5,e11=e13e2,e5,e12=2e9,e5,e14=e12,e6,e9=2e5,e6,e11=2e3,e6,e12=2e13e2,e6,e13=e9,e6,e14=e11,e7,e9=e6,e7,e12=e3,e7,e13=e1e2,e7,e14=e10,e8,e10=e7,e8,e11=e6,e8,e12=e5,e8,e13=e4,e8,e14=e12e2,e9,e10=e11,e9,e11=2e12,e9,e12=3e13,e10,e13=e14,e11,e12=3e14

(2.21)

DGsetup(LD3a, ['h1', 'h2', 'x1', 'x2', 'x3', 'x4', 'x5', 'x6', 'y1', 'y2', 'y3', 'y4', 'y5', 'y6'], ['omega']);

Lie algebra: g2a

(2.22)

 

We note that this basis is adapted to a root space decomposition.

g2a > 

RootSpaceDecomposition([h1, h2]);

table0,1=y6,3,2=y2,1,1=x3,1,0=y4,0,1=x6,1,0=x4,3,1=x5,1,1=y3,3,2=x2,2,1=y1,3,1=y5,2,1=x1

(2.23)

 

The second version is adapted to the Cartan decomposition.

g2a > 

LD3b := SimpleLieAlgebraData("g(2, Split)", g2b, version = 2);

LD3b:=e1,e2=e3,e1,e3=e2,e1,e7=14e14,e1,e8=34e11,e1,e9=32e10,e1,e10=32e9,e1,e11=3e8,e1,e12=12e13,e1,e13=12e12,e1,e14=e7,e2,e3=e1,e2,e7=34e10+12e13,e2,e8=14e13,e2,e9=12e14,e2,e10=e7