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LieAlgebras[StandardRepresentation] - find the standard matrix representation or linear vector field representation of a classical matrix algebra

Calling Sequences

     StandardRepresentation(alg, option)

     StandardRepresentation(alg, M)

Parameters

   alg      - a name or string, the name of an initialized classical matrix algebra

   M        - a name or string, the name of an initialized manifold

      option  - the keyword argument representationspace = V, where V is the name of an initialized space

 

Description

Examples

Description

• 

The first calling sequence for StandardRepresentation returns a list of matrices defining any one of the classical simple matrix algebras:

         sln, sun, sup,q, sun;

         son, sop,q, son;

         spn, ,spn,spp,q.

For convenience the following matrix algebras can also be constructed.

         gln,R, gln,C, slnn,, un, up, q, son, , soln, niln.

For the definitions and examples of all these algebras, see Details for SimpleLieAlgebraData.

• 

With the keyword argument representationspace = V, a representation is returned

• 

The second calling sequence gives the corresponding list of linear vectors fields for these algebras.

• 

The Lie algebra for these classical matrix algebras must first be created using the command SimpleLieAlgebraData.

• 

The command Query/[MatrixAlgebra] can be used to verify that a given list of matrices belongs to any one of these matrix algebras.

Examples

withDifferentialGeometry:withLieAlgebras:

 

Example 1.

We obtain the explicit matrix representation for sl3, the 8 dimensional Lie algebra of 3×3 trace-free matrices. First create the data for sl3using the SimpleLieAlgebraData command.

 

LD1SimpleLieAlgebraDatasl(3),sl3,labelformat=gl,labels=E,θ

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)

 

Initialize this Lie algebra.

DGsetupLD1

Lie algebra: sl3

(2.2)

 

Now get the explicit matrices defining sl3.

sl3 > 

A1StandardRepresentationsl3

A1:=100000001,000010001,010000000,001000000,000100000,000001000,000000100,000000010

(2.3)

 

The notation for the basis for the abstract Lie algebra sl3 was constructed to match this list of matrices:

sl3 > 

Tools:-DGinfosl3,FrameBaseVectors

E11,E22,E12,E13,E21,E23,E31,E32

(2.4)

 

The structure equations for the Lie algebra defined by the matrices A1 coincides exactly with the Lie algebra structure equations generated by the SimpleLieAlgebraData command in equation (2.1).

sl3 > 

LieAlgebraDataA1,sl3a

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

(2.5)

 

One can see by inspection that all the matrices in (2.3) are trace-free. This can also be verified using the Query command.

sl3 > 

QueryA1,sl(3),MatrixAlgebra

true

(2.6)

 

To obtain the standard vector field representation for sl3, first define a manifold with coordinates x1,x2,x3.

sl3 > 

DGsetupx1,x2,x3,V

frame name: V

(2.7)

 

We get the desired vector fields with the second calling sequence.

V > 

Γ1StandardRepresentationsl3,V

Γ1:=x1D_x1x3D_x3,x2D_x2x3D_x3,x1D_x2,x1D_x3,x2D_x1,x2D_x3,x3D_x1,x3D_x2

(2.8)

 

Again the structure equations for Γ1 are identical to those in equations (2.1) or (2.5).

V > 

LieAlgebraDataΓ1

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

(2.9)

 

Here is the standard representation, given as a representation mapping.

V > 

ρStandardRepresentationsl3,representationspace=V

ρ:=E11,100000001,E22,000010001,E12,010000000,E13,001000000,E21,000100000,E23,000001000,E31,000000100,E32,000000010

(2.10)

 

Example 2.

In this example we construct 2 different matrix representations for the Lorentz Lie algebra so3, 1. This is the 6-dimensional Lie algebra of 4×4 matrices A which are skew-symmetric with respect to a signature 3 ,1 quadratic form Q, that is, AQ + Q At = 0. There are two standard forms for Q, either

Q 1 = 0100100000100001     or     Q2= 1000010000100001,

 

which give rise to two forms for the matrices A. Either form can be generated. The default is Q1 since this form is better for calculating the Cartan subalgebra, the root space decomposition, the Cartan decomposition and so on.

V > 

LD2aSimpleLieAlgebraDataso(3, 1),so31a,labelformat=gl,labels=B,β

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,B11,B13,B14,B23,B24,B34,β11,β13,β14,β23,β24,β34

(2.11)
V > 

DGsetupLD2a

Lie algebra: so31a

(2.12)

 

Here are the defining matrices for so3,1  with respect to Q1. 

so31a > 

A2aStandardRepresentationso31a

A2a:=1000010000000000,0010000001000000,0001000000000100,0000001010000000,0000000100001000,0000000000010010

(2.13)

 

To get the alternative form for so3,1 using Q2 , add the keyword argument version = 2 to the arguments for SimpleLieAlgebraData.

V > 

LD2bSimpleLieAlgebraDataso(3,1),so31b,version=2,labelformat=gl,labels=C,γ

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,C12,C13,C23,C14,C24,C34,γ12,γ13,γ23,γ14,γ24,γ34

(2.14)
so31a > 

DGsetupLD2b

Lie algebra: so31b

(2.15)

 

Here are the defining matrices for so3,1 with respect to Q2.

so31b > 

A2bStandardRepresentationso31b

A2b:=0100100000000000,0010000010000000,0000001001000000,0001000000001000,0000000100000100,0000000000010010

(2.16)

 

We check that these matrices satisfy the defining equations AQ1 + Q1 At = 0 and AQ2 + Q2 At = 0,respectively.

so31b > 

QueryA2a,so(3,1),MatrixAlgebra

true

(2.17)
so31b > 

QueryA2b,so(3, 1),version=2,MatrixAlgebra

true

(2.18)

 

Example 3.

We give the standard representation for u3, the Lie algebra of 3×3 skew-Hermitian matrices.

so31b > 

LD3SimpleLieAlgebraDatau(3),u3,version=2,labelformat=gl,labels=S,σ

LD3:=e1,e2=e3,e1,e3=e2,e1,e4=e5,e1,e5=2e4+2e7,e1,e6=e8,e1,e7=e5,e1,e8=e6,e2,e3=e1,e2,e4=e6,e2,e5=e8,e2,e6=2e4+2e9,e2,e8=e5,e2,e9=e6,e3,e5=e6,e3,e6=e5,e3,e7=e8,e3,e8=2e7+2e9,e3,e9=e8,e4,e5=e1,e4,e6=e2,e5,e6=e3,e5,e7=e1,e5,e8=e2,e6,e8=e1,e6,e9=e2,e7,e8=e3,e8,e9=e3,S12,S13,S23,Si11,Si12,Si13,Si22,Si23,Si33,σ12,σ13,σ23,sigmai11,sigmai12,sigmai13,sigmai22,sigmai23,sigmai33

(2.19)

 

so31b > 

DGsetupLD3

Lie algebra: u3

(2.20)
u3 > 

A3StandardRepresentationu3

A3:=010100000,001000100,000001010,I00000000,0I0I00000,00I000I00,0000I0000,00000I0I0,00000000I

(2.21)

 

To calculate the structure equations for this list of matrices, as a real Lie algebra, include the keyword argument method = "real" in the calling sequence for the LieAlgebraData command.

u3 > 

LieAlgebraDataA3,u3a,method=real

e1,e2=e3,e1,e3=e2,e1,e4=e5,e1,e5=2e4+2e7,e1,e6=e8,e1,e7=e5,e1,e8=e6,e2,e3=e1,e2,e4=e6,e2,e5=e8,e2,e6=2e4+2e9,e2,e8=e5,e2,e9=e6,e3,e5=e6,e3,e6=e5,e3,e7=e8,e3,e8=2e7+2e9,e3,e9=e8,e4,e5=e1,e4,e6=e2,e5,e6=e3,e5,e7=e1,e5,e8=e2,e6,e8=e1,e6,e9=e2,e7,e8=e3,e8,e9=e3

(2.22)

We obtain the same structure equations as in (2.19).

 

We check that these matrices are skew-Hermitian but they are not all trace-free.

u3 > 

QueryA3,u(3),MatrixAlgebra

true

(2.23)
u3 > 

QueryA3,sl(3),MatrixAlgebra

false

(2.24)

 

See Also

DifferentialGeometry

LieAlgebras

LieAlgebraData

Query

Representation

SimpleLieAlgebraData