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
DGsetup(LD);
Lie algebra: so4
Use the command StandardRepresentation to retrieve the matrices for the standard representation.
M := StandardRepresentation(so4);
Define a 4-dimensional representation space V and the representation ρ.
DGsetup([x1, x2,x3,x4], V);
frame name: V
rho := Representation(so4, V, M);
Use the first calling sequence to calculate the structure equations for so4with coefficients in the representation ρ.
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
Initialize.
DGsetup(LC1);
Lie algebra with coefficients: algC1
Here is a sample calculation using a 2-form form on so4 with coefficients in V.
alpha := evalDG(x3*theta1 &w theta2);
α:=x3θ1⋀θ2
ExteriorDerivative(alpha);
−x2θ1⋀θ2⋀θ4+x4θ1⋀θ2⋀θ6+x3θ1⋀θ3⋀θ6−x3θ2⋀θ3⋀θ5
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.
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
The third calling sequence retrieves the structure equations of a previously defined Lie algebra with coefficients in a representation.
LieAlgebraWithCoefficientsData(algC1);
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