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LieAlgebras[LieAlgebraWithCoefficientsData] - calculate the structure equations for a Lie algebra with coefficients in a representation

Calling Sequences

     LieAlgebraWithCoefficientsData(ρ, rho,algC)  

     LieAlgebraWithCoefficientsData(alg,V, M, algC)

     LieAlgebraWithCoefficientsData(algC)

Parameters

   ρ       - a representation of a Lie algebra

   algC    - name or a string, the name to be assigned to the Lie algebra with coefficients

   V       - name of the representation space used to define the Lie algebra with coefficients

   M       - a list of square matrices which form a Lie algebra

Description

• 

Let V be a linear space with basis x1, x2, ... , xm ; let 𝔤 be a Lie algebra with basis e1, e2, ... , en and dual basis θ1,θ2,..., θn; and let ρ:𝔤→ glV be a representation of 𝔤. The representation ρ defines the multiplication ei⋅xa = ρeixa. Let Λp𝔤, V be the vector space of p-forms with coefficients in the representation space V. A form ω ∈Λp𝔤, Vif for all vectors X1, X2, ..., Xp ∈ 𝔤 , ωX1, X2, ..., Xp ∈V. For example, the general 1-form α and 2-form β with coefficients in V can be written as sums

α = ∑a∑i Aai xa θi     and   β= ∑a∑ijBai j xa θi ∧θj,

where the coefficients Aai and Bai jare constants. The spaces Λp𝔤, Vplay an important role in a number of constructions in Lie theory (See, for example, Cohomology, Deformation, MasseyProduct, KostantLaplacian). To work with forms defined on Lie algebras with coefficients in a representation, one first uses the commands LieAlgebraWithCoefficientsData and DGsetup -- in much the same way that one uses LieAlgebraData and AlgebraData to calculate the structure equations for a Lie algebra or a general non-commutative algebra.

• 

 The output of the LieAlgebraWithCoefficientsData is a data structure which can be passed to the command DGsetup.The structure equations are displayed.

 

See Also

DifferentialGeometry

LieAlgebras

AlgebraData

Adjoint

Cohomology

Deformation

DGsetup

LieAlgebraData

MasseyProduct

SimpleLieAlgebraData

Representation

StandardRepresentation

Examples

> 

with(DifferentialGeometry): with(LieAlgebras):

 

Example 1.

We use the 6 dimensional Lie algebra so4 and its standard representation by 4 ×4 skew-symmetric matrices to illustrate the 3 calling sequences for LieAlgebraWithCoefficientsData. First, use the command SimpleLieAlgebraData to retrieve the structure equations for so4.

> 

LD := SimpleLieAlgebraData("so(4)", so4);

LD:=e1,e2=e4,e1,e3=e5,e1,e4=−e2,e1,e5=−e3,e2,e3=e6,e2,e4=e1,e2,e6=−e3,e3,e5=e1,e3,e6=e2,e4,e5=e6,e4,e6=−e5,e5,e6=e4

(1)
> 

DGsetup(LD);

Lie algebra: so4

(2)

 

Use the command StandardRepresentation to retrieve the matrices for the standard representation.

so4 > 

M := StandardRepresentation(so4);

 

Define a 4-dimensional representation space V and the representation ρ.

so4 > 

DGsetup([x1, x2,x3,x4], V);

frame name: V

(3)
V > 

rho := Representation(so4, V, M);

 

Use the first calling sequence to calculate the structure equations for so4with coefficients in the representation ρ.

so4 > 

LC1 := LieAlgebraWithCoefficientsData(rho, algC1);

LC1:=e1,e2=e4,e1,e3=e5,e1,e4=−e2,e1,e5=−e3,e2,e3=e6,e2,e4=e1,e2,e6=−e3,e3,e5=e1,e3,e6=e2,e4,e5=e6,e4,e6=−e5,e5,e6=e4,e1.x1=x2,e1.x2=−x1,e2.x1=x3,e2.x3=−x1,e3.x1=x4,e3.x4=−x1,e4.x2=x3,e4.x3=−x2,e5.x2=x4,e5.x4=−x2,e6.x3=x4,e6.x4=−x3

(4)

 

Initialize.

so4 > 

DGsetup(LC1);

Lie algebra with coefficients: algC1

(5)

 

Here is a sample calculation using a 2-form form on so4 with coefficients in V.

algC > 

alpha := evalDG(x3*theta1 &w theta2);

α:=x3⁢θ1⁢⋀⁢θ2

(6)
algC > 

ExteriorDerivative(alpha);

−x2⁢θ1⁢⋀⁢θ2⁢⋀⁢θ4+x4⁢θ1⁢⋀⁢θ2⁢⋀⁢θ6+x3⁢θ1⁢⋀⁢θ3⁢⋀⁢θ6−x3⁢θ2⁢⋀⁢θ3⁢⋀⁢θ5

(7)

 

The second calling sequence simply allows one to calculate the structure equations (4) directly from the matrices  without having to first define the representation ρ.

so4 > 

LieAlgebraWithCoefficientsData(so4, V, M, algC2);

e1,e2=e4,e1,e3=e5,e1,e4=−e2,e1,e5=−e3,e2,e3=e6,e2,e4=e1,e2,e6=−e3,e3,e5=e1,e3,e6=e2,e4,e5=e6,e4,e6=−e5,e5,e6=e4,e1.x1=x2,e1.x2=−x1,e2.x1=x3,e2.x3=−x1,e3.x1=x4,e3.x4=−x1,e4.x2=x3,e4.x3=−x2,e5.x2=x4,e5.x4=−x2,e6.x3=x4,e6.x4=−x3

(8)

 

The third calling sequence retrieves the structure equations of a previously defined Lie algebra with coefficients in a representation.

algC > 

LieAlgebraWithCoefficientsData(algC1);

e1,e2=e4,e1,e3=e5,e1,e4=−e2,e1,e5=−e3,e2,e3=e6,e2,e4=e1,e2,e6=−e3,e3,e5=e1,e3,e6=e2,e4,e5=e6,e4,e6=−e5,e5,e6=e4,e1.x1=x2,e1.x2=−x1,e2.x1=x3,e2.x3=−x1,e3.x1=x4,e3.x4=−x1,e4.x2=x3,e4.x3=−x2,e5.x2=x4,e5.x4=−x2,e6.x3=x4,e6.x4=−x3

(9)


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