LieAlgebras[Cohomology] - compute relative Lie algebra cohomology with coefficients in a representation
LieAlgebras[RelativeChains] - find the vector space of forms on a Lie algebra relative to a given subalgebra
LieAlgebras[ CohomologyDecomposition] - decompose a closed form into the sum of an exact form and a form defining a cohomology class
Calling Sequences
RelativeChains(h)
Cohomology(C)
CohomologyDecomposition(α , H, h)
CohomologyDecomposition(α , H, R)
Parameters
h - a list of vectors in a Lie algebra 𝔤 defining a subalgebra 𝔥 ⊂𝔤
C - a list of lists C = Cp−1,Cp, Cp+1, ... , Cq+1, where Ck is a list of k-forms
α - a 𝔥 −relative, closed p−form on 𝔤
H - a list of closed p-forms on 𝔤 defining the basis for the (relative) cohomology of 𝔤 in degree p
R - a list of (p-1)-forms on 𝔤 defining the basis for the relative chains of 𝔤 in degree p−1
Description
Examples
Let 𝔤 be a n-dimensional (real) Lie algebra. Let 𝔤* be the dual space of 𝔤 (the space of 1-forms on 𝔤). When initializing a Lie algebra with DGsetup, the default labelling is e1, e2, ..., en for the basis vectors and θ1, θ2, ..., θn for the 1-forms. Denote by Λp𝔤* the p−forms on 𝔤 : these are the alternating mult-linear maps ω : 𝔤 × 𝔤 ⋅⋅⋅ × 𝔤 → ℝ. Let ρ: 𝔤 → glV be a representation of 𝔤 . If x1, x2, ... xm is a basis for V, let ρeixα = Bα iβxβ and denote by Λp𝔤*, V the p−forms on 𝔤 with coefficients in V. These are the alternating mult-linear maps ω : 𝔤 × 𝔤 ⋅⋅⋅ × 𝔤 → V. Any form ω ∈ Λpg*, V can be written as
ω = Ai1i2⋅⋅⋅ipα xα θi1∧ θi2∧ ⋅⋅⋅ ∧θip.
The exterior derivative d: Λpg*, V→Λp+1g*, V is defined by the rules dθi = −12 Cjki θj ∧θk and dxα = Bα iβxβθi. If 𝔥 ⊂𝔤 is a subalgebra of 𝔤, then the space of 𝔥−relative p−forms on 𝔤 with coefficients in V is
Λp𝔤*, 𝔥, V= {ω ∈Λpg*, V | ιyω =0 and ιydω =0 for all y ∈ 𝔥 } .
A p-form ω ∈Λp𝔤*, 𝔥, Vis closed if dω = 0 and exact if there a p −1-form η ∈Λp−1𝔤*, hfr,V such that ω = dη. The 𝔥−relative ,p-dimensional Lie algebra cohomology of 𝔤 with coefficients in the representation V is the space of closed p -forms module the exact p-forms, that is,
Hp𝔤, 𝔥, V = {ω ∈ Λp𝔤*, 𝔥, V | dω = 0}{ω ∈ Λp𝔤*, 𝔥, V| ω = dη} .
The cohomology Hp𝔤 of 𝔤, the relative Lie algebra cohomology Hp𝔤, 𝔥, and the cohomology Hp𝔤, Vof 𝔤 with coefficients in a represention all play an important role in Lie theory, in the differential geometry and topology of homogeneous spaces and in the Cartan equivalence method. The text by D. B. Fuks (Chapter 1) and the papers by Hochschild and Koszul contain the basic material on Lie Algebra cohomology. Also, see the help pages Deformation, Extensions, KostantCodifferential.
The LieAlgebra package currently contains 3 commands: RelativeChains, Cohomology, and CohomologyDecomposition for finding Lie algebra cohomology.
The command RelativeChains(h) returns a list C = C1,C2, C3 ... of all relative chains Λp𝔤*, 𝔥 , V.
The command Cohomology(C) computes the cohomology of the sequence of forms C = Cp−1,Cp, Cp+1 ,..., Cq+1. This requires that dCi ⊂ Ci+1 for all i =p−1, p , p +1, ..., q. If Cp−1 is a list of p−1 forms on 𝔤, then Cohomology(C) returns a list H= Hp, Hp+1, ... , Hq, where Hk is a basis for the cohomology in Ck.
The command CohomologyDecomposition(alpha, H, h) returns a pair of forms β, δ such that α = β + dδ, where β is a linear combination of the cohomology representatives given by H and where δ is a 𝔥−relative form The form β is uniquely determined, the form δ is not. In particular, if the closed form α is exact, then β = 0.
withDifferentialGeometry:withLieAlgebras:
Example 1.
First we initialize a Lie algebra.
L1≔_DGLieAlgebra,Alg1,5,2,3,1,1,2,5,3,1,4,5,4,1
L1≔e2,e3=e1,e2,e5=e3,e4,e5=e4
DGsetupL1
Lie algebra: Alg1
For this example we take h to be the trivial subspace. In this case the procedure RelativeChains simply returns a list of bases for the 1-forms on g, the 2-forms on g, the 3-forms on g, and so on.
C≔RelativeChains
C≔,θ1,θ2,θ3,θ4,θ5,θ1⋀θ2,θ1⋀θ3,θ1⋀θ4,θ1⋀θ5,θ2⋀θ3,θ2⋀θ4,θ2⋀θ5,θ3⋀θ4,θ3⋀θ5,θ4⋀θ5,θ1⋀θ2⋀θ3,θ1⋀θ2⋀θ4,θ1⋀θ2⋀θ5,θ1⋀θ3⋀θ4,θ1⋀θ3⋀θ5,θ1⋀θ4⋀θ5,θ2⋀θ3⋀θ4,θ2⋀θ3⋀θ5,θ2⋀θ4⋀θ5,θ3⋀θ4⋀θ5,θ1⋀θ2⋀θ3⋀θ4,θ1⋀θ2⋀θ3⋀θ5,θ1⋀θ2⋀θ4⋀θ5,θ1⋀θ3⋀θ4⋀θ5,θ2⋀θ3⋀θ4⋀θ5,θ1⋀θ2⋀θ3⋀θ4⋀θ5,
We pass the output of the RelativeChains program to the Cohomology program.
H≔CohomologyC
H≔θ5,θ2,θ1⋀θ2,θ3⋀θ5,θ1⋀θ2⋀θ3,θ1⋀θ3⋀θ5,θ1⋀θ2⋀θ3⋀θ5,
To read off the dimensions of the cohomology of g, use the nops and map command.
mapnops,H
2,2,2,1,0
Example 2.
We continue with Example 1. To find the cohomology of 𝔤 just in degree 3, pass the Cohomology program to just the chains of degree 2 and 3 and 4.
CohomologyC3..5
θ1⋀θ2⋀θ3,θ1⋀θ3⋀θ5
Example 3.
We continue with Example 1. Show that the 2-form β is closed and express β as a linear combination of the cohomology classes in H2 and the exterior derivative of a 1-form.
α≔evalDGθ4&wθ5−θ3&wθ5+3θ2&wedgeθ5+2θ1&wθ2
α≔2θ1⋀θ2+3θ2⋀θ5−θ3⋀θ5+θ4⋀θ5
ExteriorDerivativeα
0θ1⋀θ2⋀θ3
β,δ≔CohomologyDecompositionα,H2
β,δ≔2θ1⋀θ2−θ3⋀θ5,−3θ3−θ4
α&minusβ&plusExteriorDerivativeδ
0θ1⋀θ2
Example 4.
L2≔_DGLieAlgebra,Alg2,5,2,3,1,1,2,5,3,1,4,5,4,1
L2≔e2,e3=e1,e2,e5=e3,e4,e5=e4
DGsetupL2
Lie algebra: Alg2
Define a 2 dimensional subspace h to be the vectors spanned by S..
S≔e1,e2
Compute the relative chains with respect to the subspace h.
C≔RelativeChainsS
C≔,θ4,θ5,−θ3⋀θ5,−θ4⋀θ5,−θ3⋀θ4⋀θ5,
H≔θ5,−θ3⋀θ5,−θ3⋀θ4⋀θ5
Example 5.
In this example we compute the cohomology of a 4-dimensional Lie algebra with coefficients in the adjoint representation. First define and initialize the Lie algebra.
L3≔Library:-RetrieveWinternitz,1,4,7,Alg3
L3≔e1,e4=2e1,e2,e3=e1,e2,e4=e2,e3,e4=e2+e3
DGsetupL3
Lie algebra: Alg3
Define the representation space V.
DGsetupx1,x2,x3,x4,V
frame name: V
Define the adjoint representation.
ρ≔RepresentationAlg3,V,AdjointAlg3
ρ≔e1,0002000000000000,e2,0010000100000000,e3,0−100000100010000,e4,−20000−1−1000−100000
DGsetupAlg3,ρ,Rep1
Lie algebra with coefficients: Rep1
Note that the chains are now linear functions of the coordinates on the representation space.
C≔x1,x2,x3,x4,x1θ1,x1θ2,x1θ3,x1θ4,x2θ1,x2θ2,x2θ3,x2θ4,x3θ1,x3θ2,x3θ3,x3θ4,x4θ1,x4θ2,x4θ3,x4θ4,x1θ1⋀θ2,x1θ1⋀θ3,x1θ1⋀θ4,x1θ2⋀θ3,x1θ2⋀θ4,x1θ3⋀θ4,x2θ1⋀θ2,x2θ1⋀θ3,x2θ1⋀θ4,x2θ2⋀θ3,x2θ2⋀θ4,x2θ3⋀θ4,x3θ1⋀θ2,x3θ1⋀θ3,x3θ1⋀θ4,x3θ2⋀θ3,x3θ2⋀θ4,x3θ3⋀θ4,x4θ1⋀θ2,x4θ1⋀θ3,x4θ1⋀θ4,x4θ2⋀θ3,x4θ2⋀θ4,x4θ3⋀θ4,x1θ1⋀θ2⋀θ3,x1θ1⋀θ2⋀θ4,x1θ1⋀θ3⋀θ4,x1θ2⋀θ3⋀θ4,x2θ1⋀θ2⋀θ3,x2θ1⋀θ2⋀θ4,x2θ1⋀θ3⋀θ4,x2θ2⋀θ3⋀θ4,x3θ1⋀θ2⋀θ3,x3θ1⋀θ2⋀θ4,x3θ1⋀θ3⋀θ4,x3θ2⋀θ3⋀θ4,x4θ1⋀θ2⋀θ3,x4θ1⋀θ2⋀θ4,x4θ1⋀θ3⋀θ4,x4θ2⋀θ3⋀θ4,x1θ1⋀θ2⋀θ3⋀θ4,x2θ1⋀θ2⋀θ3⋀θ4,x3θ1⋀θ2⋀θ3⋀θ4,x4θ1⋀θ2⋀θ3⋀θ4,
CohomologyC
−x1θ13−x2θ26+−x36+x2θ3,x3θ2⋀θ4,,
Example 6.
Finally, we compute the Lie algebra cohomology of Alg3 with coefficients in the adjoint representation, relative to the subalgebra spanned bu e1.
C≔RelativeChainse1
C≔x3,x2,x1,x3θ2,x3θ3,x3θ4,x2θ2,x2θ3,x2θ4,x1θ2,x1θ3,x1θ4,−x3θ3⋀θ4,−x2θ2⋀θ3,−x2θ2⋀θ4,−x2θ3⋀θ4,−x1θ2⋀θ3,−x1θ2⋀θ4,−x1θ3⋀θ4,−x3θ2⋀θ3,−x3θ2⋀θ4,−x3θ2⋀θ3⋀θ4,−x2θ2⋀θ3⋀θ4,−x1θ2⋀θ3⋀θ4,
x2θ3,−x3θ2⋀θ4,
See Also
DifferentialGeometry
LieAlgebras
Deformation
ExteriorDerivative
Extension
KostantCodifferential
KostantLaplacian
MasseyProduct
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