LieAlgebras[KostantCodifferential] - calculate the Kostant co-differential of a p-form or a list of p-forms defined on a nilpotent Lie algebra with coefficients in a representation
LieAlgebras[KostantLaplacian] - calculate the Kostant Laplacian of a form defined on a nilpotent Lie algebra with coefficients in a representation
Calling Sequences
KonstantCodifferential(ω, B,invB)
KostantLaplacian(ω,B,invB, method)
Parameters
ω - a form defined on a nilpotent Lie algebra with coefficients in a representation V
B - (optional) the Killing form for the associated semi-simple Lie algebra
invB - (optional) the inverse of the Killing form
method - with the keyword argument method = "LieDerivative", the Kostant Laplacian is computed in terms of the Lie derivative operator
Description
Examples
Let 𝔤 be a semi-simple Lie algebra and let V be a representation space for 𝔤. Fix a grading of 𝔤 :
𝔤 = 𝔤−k ⊕ 𝔤−k+1⊕⋅⋅⋅⊕𝔤0⋅⋅⋅⊕𝔤k −1⊕𝔤k and let 𝔤−= 𝔤−k ⊕ 𝔤−k+1⊕⋅⋅⋅⊕𝔤−1 and 𝔭= 𝔤0⊕⋅⋅⋅⊕ 𝔤k −1⊕𝔤k .
Such gradings can be systematically constructed using the command GradeSemiSimpleLieAlgebra. The Lie algebra 𝔤− is nilpotent and the representation V for 𝔤 restricts to give a representation space for 𝔤−. The commands Representation and RestrictedRepresentation can be used to create these representations for 𝔤 and 𝔤−.
Now let Λp𝔤−, V denote the space of p-forms on 𝔤− with coefficients in V. If ω ∈ Λp𝔤−, V and X1, ..., Xp are vectors in 𝔤−, then ω X1, ..., Xp ∈ V. The spaces Λp𝔤−, V can be constructed using the command DGsetup and RelativeChains. The exterior derivative defines a mapping d :Λp𝔤−, V → Λp+1𝔤−, V while the Kostant co-differential is a map d*:Λp𝔤−, V → Λp−1𝔤−, V.
The Kostant co-differential is easily defined in terms of the general Codifferential operator ∂. Let α ∈ Λp𝔤−, V and let X=α+be the degree p multi-vector obtained by raising the indices of the form α using the inverse of the Killing metric for 𝔤. Take the co-differential of X to obtain the multi-vector ∂X of degree p−1 and use the Killing form to lower the indices of ∂X to obtain a form. This is the Kostant co-differential of α, that is,
d*α = ∂α+−.
The command PositiveDefiniteMetricOnRepresentationSpace can be to used to construct positive definite inner products on 𝔤 and on V such that the Kostant co-differential is the adjoint of the exterior derivative operator with respect to the induced inner product ⋅, ⋅ on Λp𝔤−, V, that is,
d*α, β = α, dβ for all α ϵ Λp𝔤−, V and β ϵ Λp−1𝔤−, V.
The Kostant Laplacian is the map Δ:Λp𝔤−, V → Λp𝔤−, V defined by
Δ = d d* + d*d .
The Kostant co-differential and Kostant Laplacian are very useful for the explicit calculation of the Lie algebra cohomology of the complex Λp𝔤−, V as
Hp𝔤−, V = ker d ∩ ker d* = ker Δ
This is generally a much faster way to calculate these cohomology spaces than with a direct calculation using the command Cohomology. See also the command DGNullSpace.
The optional arguments (Killing form and inverse Killing form) for KostantCodifferential and KostantLaplacian dramatically improve the computational efficiency of these commands.
The formula for the Kostant Laplacian in terms of the Lie derivative is as follows. First pick a basis ζ𝓁 for 𝔤 which is adapted to the decomposition 𝔤 = 𝔤− ⊕ 𝔭. Let ηk be the dual basis for defined with respect to the Killing form B, that is, B(ηk , ζ𝓁 )=δk 𝓁. . Then
Δ = −12∑ℓ Lη𝓁V Lζ𝓁V − 12 ∑𝓁: ζ𝓁 ϵ 𝔤−Lη𝓁 Lζ𝓁 + 12 ∑𝓁: ζ𝓁 ϵ 𝔭Lη𝓁 Lζ𝓁 .
While this formula has important theoretical implications, it is used here primary as a consistency check on the software implementation of the Kostant Laplacian.
restart:withDifferentialGeometry:withTensor:withLieAlgebras:
In this series of examples and applications we shall work with a 3-step graduation of the 10-dimensional real symplectic Lie algebra sp4, Rwith coefficients in the adjoint representation.The following steps are needed to create the environment for defining the Kostant codifferential and the Kostant Laplacian.
Use the command SimpleLieAlgebraData and DGsetup to initialize a simple Lie algebra 𝔤.
Use the command SimpleLieAlgebraProperties to retrieve the simple roots of 𝔤 . Use the command GradeSemiSimpleAlgebra to construct a gradation of 𝔤.
Initialize the Lie algebras 𝔤 and 𝔤−.
Use the commands StandardRepresentation, Adjoint, Representation etc. to make a representation V of 𝔤 . Use the RestrictedRepresentation command to restrict the representation of 𝔤 on V to 𝔤−
Initialize the Lie algebra of 𝔤 with coefficients in V. Initialize the Lie algebra of 𝔤− with coefficients in V.
Step 1. Use the command SimpleLieAlgebraData to retrieve the structure equations for sp4, R. Initialize this Lie algebra.
LD≔SimpleLieAlgebraDatasp(4, R),alg
LD≔e1,e2=e2,e1,e3=−e3,e1,e5=2e5,e1,e6=e6,e1,e8=−2e8,e1,e9=−e9,e2,e3=e1−e4,e2,e4=e2,e2,e6=2e5,e2,e7=e6,e2,e8=−e9,e2,e9=−2e10,e3,e4=−e3,e3,e5=e6,e3,e6=2e7,e3,e9=−2e8,e3,e10=−e9,e4,e6=e6,e4,e7=2e7,e4,e9=−e9,e4,e10=−2e10,e5,e8=e1,e5,e9=e2,e6,e8=e3,e6,e9=e1+e4,e6,e10=e2,e7,e9=e3,e7,e10=e4
DGsetupLD,U,υ
Lie algebra: alg
Step 2. Various properties of the classical Lie algebras are available with the command SimpleLieAlgebraProperties. We need the simple roots here.
Properties≔SimpleLieAlgebraPropertiesalg:
Δ0≔PropertiesSimpleRoots
Δ0≔1−1,02
Every subset of the simple roots of a Lie algebra defines a grading of that algebra. Here we use all the roots of sp4, R to obtain a 2-step gradation with the command GradeSemiSimpleLieAlgebra.
G≔GradeSemiSimpleLieAlgebraΔ0,Properties
G≔table−1=U3,U10,0=U1,U4,−2=U9,1=U2,U7,−3=U8,2=U6,3=U5
Step 3. Note that the vectors U8, U9, U3, U10 define the negative part of sp4, R with respect to the chosen grading. The next step is to introduce a new basis for alg adapted to the grading. We call sp4, R in this new basis sp4 and we call the negatively graded part N.
The negatively graded component N is always nilpotent. With the following calling sequence LieAlgebraData returns the structure equations for sp4, the structure equations for N, and the basis of our original algebra, adapted to the grading, and a basis for N.
LD1,LD2,B1,B2≔LieAlgebraDataG,sp4,negative,N,output=basis
LD1,LD2,B1,B2≔e1,e5=2e1,e1,e7=e2,e1,e9=−e3,e1,e10=−e5,e2,e3=2e1,e2,e5=e2,e2,e6=e2,e2,e7=2e4,e2,e8=−e3,e2,e9=−e6−e5,e2,e10=−e7,e3,e4=−e2,e3,e5=e3,e3,e6=−e3,e3,e7=e6−e5,e3,e9=2e8,e3,e10=e9,e4,e6=2e4,e4,e8=−e6,e4,e9=−e7,e5,e7=e7,e5,e9=e9,e5,e10=2e10,e6,e7=−e7,e6,e8=2e8,e6,e9=e9,e7,e8=e9,e7,e9=2e10,e2,e3=2e1,e3,e4=−e2,U8,U9,U3,U10,U1,U4,U2,U7,U6,U5,U8,U9,U3,U10
Initialize the Lie algebras sp4 and N.
DGsetupLD1
Lie algebra: sp4
DGsetupLD2
Lie algebra: N
Step 4. Now we are ready to define the adjoint representation for sp4 and its restriction to N. Since sp4 is 10-dimensional, we need a 10-dimensional representation space. Call it V.
DGsetupx1,x2,x3,x4,x5,x6,x7,x8,x9,x10,V,grading=−3,−2,−1,−1,0,0,1,1,2,3
frame name: V
The command Adjoint gives the adjoint representation for sp4.
ρ1≔Adjointsp4,representationspace=V
ρ1≔e1,0000200000000000100000000000−100000000000000000000−100000000000000000000000000000000000000000000000000,e2,002000000000001100000000000−100000000200000000000−1000000000−10000000000−1000000000000000000000000000000,e3,0−200000000000−100000000001−100000000000000000000−100000000010000000000000000000002000000000010000000000,e4,000000000000100000000000000000000002000000000000000000000−10000000000−10000000000000000000000000000000,e5,−20000000000−10000000000−100000000000000000000000000000000000000000001000000000000000000000100000000002,e6,00000000000−1000000000010000000000−200000000000000000000000000000000−1000000000020000000000100000000000,e7,0000000000−100000000000000000000−200000000001000000000−100000000000−110000000000000000000001000000000020,e8,000000000000000000000100000000000000000000000000000001000000000000000000000−20000000000−10000000000000,e9,000000000000000000001000000000000000000001000000000100000000000100000000−200000000000−1−10000000000−2000,e10,0000000000000000000000000000000000000000100000000000000000000100000000000000000000−100000000000−200000
The command RestrictedRepresentation gives the restriction of the adjoint representation for sp4, R to the subalgebra N. (The following calling sequence assumes that the first 4 vectors in the given basis define the subalgebra N. )
ρ2≔RestrictedRepresentationρ1,N
ρ2≔e1,0000200000000000100000000000−100000000000000000000−100000000000000000000000000000000000000000000000000,e2,002000000000001100000000000−100000000200000000000−1000000000−10000000000−1000000000000000000000000000000,e3,0−200000000000−100000000001−100000000000000000000−100000000010000000000000000000002000000000010000000000,e4,000000000000100000000000000000000002000000000000000000000−10000000000−10000000000000000000000000000000
Step 5. Initialize the Lie algebra sp4, R with coefficients in the adjoint representation. Call it sp4V and label the basis vectors X1, ... , X8 and the basis 1-forms ξ1, ... , ξ8.
DGsetupρ1,sp4V,X,ξ
Lie algebra with coefficients: sp4V
Initialize the Lie algebra N with coefficients in the adjoint representation of sp4, R. Call it NV and label the basis vectors O1, ... , O4 and the basis 1-forms o1, ... ,o4. As described above the Kostant codifferential uses the embedding of N in sp4V. This information is provided by the keyword argument ambientalgebra.
DGsetupρ2,NV,O,o,ambientalgebra=sp4V
Lie algebra with coefficients: NV
Example 1.
Here are some sample calculations of the Kostant co-differential.
α≔evalDGx1o2
α≔x1o2
KostantCodifferentialα
x312
β≔evalDGx3o2&wo3
β≔x3o2⋀o3
KostantCodifferentialβ
−x312o1−x512−x612o2−x86o3
Let's do this last calculation directly from the definition. For this we need the Killing form and its inverse.
B≔KillingFormsp4V
B≔6ξ1ξ10+12ξ2ξ9+12ξ3ξ7+6ξ4ξ8+12ξ5ξ5+12ξ6ξ6+12ξ7ξ3+6ξ8ξ4+12ξ9ξ2+6ξ10ξ1
invB≔InverseMetricB
invB≔16X1X10+112X2X9+112X3X7+16X4X8+112X5X5+112X6X6+112X7X3+16X8X4+112X9X2+16X10X1
Re-define β as a form on sp4R.
β1≔evalDGx3ξ2&wξ3
β1≔x3ξ2⋀ξ3
Use the inverse of the Killing form to convert β to a multi-vector X:
X≔RaiseLowerIndicesinvB,β1,1,2
X≔−x3144X7⋀X9
Take the co-differential of X.
Y≔CodifferentialX
Y≔−x872X7−x5144−x6144X9−x372X10
Lower the indices of Y with the Killing form.
RaiseLowerIndicesB,Y,1
−x312ξ1−x512−x612ξ2−x86ξ3
Example 2.
The square of the Kostant co-differential vanishes.
α2≔evalDGx3o1&wo2&wo3
α2≔x3o1⋀o2⋀o3
β2≔KostantCodifferentialβ
β2≔−x312o1−x512−x612o2−x86o3
KostantCodifferentialβ2
0
Example 3.
We check that the Kostant co-differential is the adjoint of the exterior derivative. Here are the inner products we need (See PositiveDefiniteMetricOnRepresentationSpace) .
g≔evalDG6o1&to1+12o2&to2+12o3&to3+6o4&to4
g≔6o1o1+12o2o2+12o3o3+6o4o4
gV≔evalDGdx1&tdx1+2dx2&tdx2+2dx3&tdx3+dx4&tdx4+2dx5&tdx5+2dx6&tdx6+2dx7&tdx7+dx8&tdx8+2dx9&tdx9+dx10&tdx10
gV≔dx1dx1+2dx2dx2+2dx3dx3+dx4dx4+2dx5dx5+2dx6dx6+2dx7dx7+dx8dx8+2dx9dx9+dx10dx10
α3≔evalDGx3o1&wo2
α3≔x3o1⋀o2
β3≔evalDGx8&wedgeo1
β3≔x8o1
Here is the left-hand side of the adjoint equation.
FormInnerProductg,gV,KostantCodifferentialα3,β3
136
Here is the right-hand side of the adjoint equation.
FormInnerProductg,gV,α3,ExteriorDerivativeβ3
We can easily check the adjoint equation for lists of forms. We use the command RelativeChains to generate lists of forms. For this example, we specify the weight of the forms to keep the lists small.
ChangeFrameNV
V
Λ1≔RelativeChains,1,weight=1
Λ1≔x2o1,x3o2,x4o2,x5o3,x5o4,x6o3,x6o4
Λ2≔RelativeChains,2,weight=1
Λ2≔x1o1⋀o3,x1o1⋀o4,x2o2⋀o3,x2o2⋀o4,x3o3⋀o4,x4o3⋀o4
LHS≔FormInnerProductg,gV,KostantCodifferentialΛ2,Λ1
LHS≔13600136000000011800−1360172172017200−1360013601360136001360−1360017200−1360
RHS≔FormInnerProductg,gV,Λ2,ExteriorDerivativeΛ1
RHS≔13600136000000011800−1360172172017200−1360013601360136001360−1360017200−1360
The equality of these matrices verifies the adjoint equation for given lists of forms.
LHS−RHS
000000000000000000000000000000000000000000
Example 4.
Here are some sample calculations of the Kostant Laplacian.
A scalar:
α4≔x6
KostantLaplacianα4
x62
A 2-form:
β4≔evalDGx3o1&wo4
β4≔x3o1⋀o4
KostantLaplacianβ4
x32o1⋀o4
A 3-form:
δ4≔evalDGx3o1&wo3&wo4
δ4≔x3o1⋀o3⋀o4
KostantLaplacianδ4
2x33o1⋀o3⋀o4
Example 5.
We calculate the second cohomology H2𝔤−, V.
NV
Here are the 2-chains Λ2𝔤−, V.
Λ2≔RelativeChains,2
Λ2≔x1o1⋀o2,x1o1⋀o3,x1o1⋀o4,x1o2⋀o3,x1o2⋀o4,x1o3⋀o4,x2o1⋀o2,x2o1⋀o3,x2o1⋀o4,x2o2⋀o3,x2o2⋀o4,x2o3⋀o4,x3o1⋀o2,x3o1⋀o3,x3o1⋀o4,x3o2⋀o3,x3o2⋀o4,x3o3⋀o4,x4o1⋀o2,x4o1⋀o3,x4o1⋀o4,x4o2⋀o3,x4o2⋀o4,x4o3⋀o4,x5o1⋀o2,x5o1⋀o3,x5o1⋀o4,x5o2⋀o3,x5o2⋀o4,x5o3⋀o4,x6o1⋀o2,x6o1⋀o3,x6o1⋀o4,x6o2⋀o3,x6o2⋀o4,x6o3⋀o4,x7o1⋀o2,x7o1⋀o3,x7o1⋀o4,x7o2⋀o3,x7o2⋀o4,x7o3⋀o4,x8o1⋀o2,x8o1⋀o3,x8o1⋀o4,x8o2⋀o3,x8o2⋀o4,x8o3⋀o4,x9o1⋀o2,x9o1⋀o3,x9o1⋀o4,x9o2⋀o3,x9o2⋀o4,x9o3⋀o4,x10o1⋀o2,x10o1⋀o3,x10o1⋀o4,x10o2⋀o3,x10o2⋀o4,x10o3⋀o4
For these cohomology calculations it is preferable to specify the Killing form and its inverse in the arguments for KostantCodifferential and KostantLaplacian. Otherwise the Killing form and its inverse will be re-calculated for each 2 form in Λ2𝔤−, V.
Method 1. Here we shall calculate H2𝔤−, V = ker d ∩ ker d* using the DGNullSpace and IntersectSubspaces commands.
Here is ker d .
KerD≔DGNullSpaceExteriorDerivative,Λ2,method=real
KerD≔x1o1⋀o2+−x62+x52o3⋀o4,x1o1⋀o3,x1o1⋀o4+x3o3⋀o4,x1o2⋀o3,x1o2⋀o4,x1o3⋀o4,x2o1⋀o2+x5−x6o2⋀o3+x7o3⋀o4,x2o1⋀o3,x2o1⋀o4−−3x6+x5o3⋀o4,x2o2⋀o3,x2o2⋀o4−x3o3⋀o4,x2o3⋀o4,x3o1⋀o2+2x8o2⋀o3−x7o2⋀o4−x9o3⋀o4,x3o1⋀o3+x5+x6o2⋀o3−x7o3⋀o4,x3o1⋀o4+x5o2⋀o4−3x8o3⋀o4,x3o2⋀o3−x6o3⋀o4,x3o2⋀o4,x4o1⋀o2+x62o1⋀o3+3x72o2⋀o3,x4o1⋀o3,x4o1⋀o4+2x6o2⋀o3−x7o3⋀o4,x4o2⋀o3,x4o2⋀o4−x6o3⋀o4,x4o3⋀o4,x5o1⋀o2+x9o2⋀o3−x10o3⋀o4,x5o1⋀o3+x7o2⋀o3,x5o1⋀o4+x7o2⋀o4−x9o3⋀o4,x6o1⋀o2−2x8o1⋀o3+x7o1⋀o4−3x9o2⋀o3+x10o3⋀o4,x6o1⋀o4−4x8o2⋀o3+x7o2⋀o4+x9o3⋀o4,x6o2⋀o4+x8o3⋀o4,x7o1⋀o2−x9o1⋀o3−2x10o2⋀o3,x8o2⋀o4
Here is ker d*. First we make a small procedure which fixes the Killing form and its inverse in the arguments for KostantCodifferential.
KCoD≔θ↦KostantCodifferentialθ,B,invB
KCoD≔θ↦LieAlgebras:−KostantCodifferentialθ,B,invB
KerKCoD≔DGNullSpaceKCoD,Λ2,method=real
KerKCoD≔x1o1⋀o2−x3o2⋀o3−x4o2⋀o4−x62+3x52o3⋀o4,x1o1⋀o4−x2o2⋀o4−x3o3⋀o4,x2o1⋀o2−x5+x6o2⋀o3−x7o3⋀o4,x2o1⋀o4−2x3o2⋀o3−2x4o2⋀o4−x5+x6o3⋀o4,x3o1⋀o2+2x8o2⋀o3+x7o2⋀o4+x9o3⋀o4,x3o1⋀o3−x5−x6o2⋀o3+x7o3⋀o4,x3o1⋀o4−2x63o2⋀o4+4x83o3⋀o4,x4o1⋀o2−x6o1⋀o3,x4o1⋀o3,x4o1⋀o4−2x6o2⋀o3−x7o3⋀o4,x5o1⋀o2+x9o2⋀o3+3x10o3⋀o4,3x6+x5o1⋀o3−2x7o2⋀o3,x5o1⋀o4−x7o2⋀o4+x9o3⋀o4,−x63+x5o2⋀o4+2x83o3⋀o4,x6o1⋀o2+x9o2⋀o3+x10o3⋀o4,x6o1⋀o4+4x8o2⋀o3+x7o2⋀o4+x9o3⋀o4,x7o1⋀o2+2x10o2⋀o3,x7o1⋀o3,x7o1⋀o4+2x9o2⋀o3+2x10o3⋀o4,x8o1⋀o2,x8o1⋀o3−x9o2⋀o3−x10o3⋀o4,x8o1⋀o4,x8o2⋀o4,x9o1⋀o2,x9o1⋀o3−2x10o2⋀o3,x9o1⋀o4,x9o2⋀o4,x10o1⋀o2,x10o1⋀o3,x10o1⋀o4,x10o2⋀o4
Now calculate the intersection ker d ∩ ker d*.
IntersectSubspacesKerD,KerKCoD,method=real
x4o1⋀o3,x8o2⋀o4
Method 2. Here we shall calculate H2𝔤−, V = ker Δ. Again we make a small procedure which fixes the Killing form and its inverse in the arguments for KostantLaplacian.
KBox≔θ↦KostantLaplacianθ,B,invB
KBox≔θ↦LieAlgebras:−KostantLaplacianθ,B,invB
DGNullSpaceKBox,Λ2,method=real
See Also
Adjoint
Cohomology
DGNullSpace
ExteriorDerivative
GradeSemiSimpleLieAlgebra
FormInnerProduct
IntersectSubspaces
LieAlgebraData
PositiveDefiniteMetricOnRepresentationSpace
RestrictedRepresentation
SimpleLieAlgebraData
SimpleLieAlgebraProperties
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