LieAlgebras[Deformation] - find the deformation of a Lie algebra defined by a list of 2-forms
Calling Sequences
Deformation(Alg,Ω,t, AlgName,option)
Parameters
Alg - the name of an initialized Lie algebra 𝔤
Ω − a list of 2-forms on 𝔤 with values in 𝔤
t - an unassigned name to be used as the deformation parameter, or a list of unassigned names
AlgName - an unassigned name (or string) for the deformation algebra
option - the keyword argument parameters = [a, b, ... ]
Description
Examples
Let 𝔤 be a finite-dimensional Lie algebra. A deformation of 𝔤 is a smoothly varying family of Lie algebras 𝔤t (all of the same dimension) such that 𝔤0 = 𝔤. The deformation is called trivial if the Lie algebras 𝔤t are isomorphic for all values of t. Deformations are calculated as a formal power series for the bracket operation in 𝔤t
x, y t= x,y +t η1x, y +t2 η2x, y + t3 η3x, y + t4η4x, y +⋅⋅⋅
Here x,y ∈ 𝔤 and each coefficient ηk is a bilinear, skew-symmetric mapping ηk : 𝔤 × 𝔤 →𝔤, that is, ηk ∈ Λ2𝔤, 𝔤. The Jacobi identity for the bracket ⋅,⋅ t imposes a set of conditions on the coefficients η1, η2 , η3, ... . These conditions are described below in equations (1), (2) and (3).
The command Deformation will return the structure equations for the bracket operation x, yt using the Lie bracket x,y defined by the first argument Alg and the forms Ω =η1, η2 , η3, ... given by the second argument. The procedure Deformation does not verify that the forms η1, η2 , η3,satisfy the conditions (1), (2) and (3) below so that the bracket operation x, yt need not satisfy the Jacobi identity.
Suppose that the forms Ω =η1, η2 , η3, ... depend upon a number of parameters a, b, ... .With the keyword argument parameters = [a, b, ...], the procedure Deformation initializes the deformation algebra defined by x, yt (using the name provided by the 4th argument) and calculates the conditions on these parameters imposed by the Jacobi identities. A sequence TF, Eq, Soln, LD of 4 elements is returned, where TF is true if there is a set of parameter values satisfying the Jacobi identities, Eq is the set of equations arising from the Jacobi equations, Soln is the list of solutions to the Jacobi equations for the parameters a, b, ... and LD the Lie algebra data structures defined by these solutions.
The conditions imposed on the coefficients η1, η2 , η3, ... by the Jacobi identity for the bracket ⋅,⋅ t are as follows. First, the 2-form η1 must be closed, that is,
dη1 = 0, (1)
where d is the exterior derivative operator. If η1 is an exact form, that is, η1 = dξ1, then the linear deformation x, y t= x,y +t η1x, y is a trivial deformation. Thus, to determine the possible non-trivial deformations, one first computes the cohomology H2𝔤, 𝔤. This can be done with the commands Representation, RelativeChains, Cohomology.
If the Massey product η1,η1 of the linear deformation η1 vanishes, then the Jacobi identity holds and the linear deformation
x, y t = x,y +t η1x, y
defines a Lie algebra. Otherwise, the quadratic deformation η2 can be determined by the equation
dη2 + η1,η1 = 0. (2)
This implies that for the quadratic deformation to exist, the Massey product η1,η1 must be an exact 2-form. The quadratic deformation can be found using the command CohomologyDecomposition. The higher order deformations are determined by the equations
dη3 +η1,η2 = 0, dη4 + η1,η3+ η3, η1+ η2, η2 = 0, etc. (3)
See D. B. Fuks Cohomology of Infinite Dimensional Lie Algebras (pages 35 - 38) for more details on deformations of Lie algebras and other applications of Lie algebra cohomology.
withDifferentialGeometry:withLieAlgebras:
First initialize an 8-dimensional Lie algebra. We shall create various deformations of this Lie algebra. Here are the structure equations.
StrEq≔x4,x6=−x3,x4,x7=−x1,x5,x6=−x1,x5,x7=−x2,x6,x8=−x4,x7,x8=−x5
StrEq:=x4,x6=−x3,x4,x7=−x1,x5,x6=−x1,x5,x7=−x2,x6,x8=−x4,x7,x8=−x5
Use the commands LieAlgebraData and DGsetup to initialize this Lie algebra.
LD≔LieAlgebraDataStrEq,x1,x2,x3,x4,x5,x6,x7,x8,alg
LD:=e4,e6=−e3,e4,e7=−e1,e5,e6=−e1,e5,e7=−e2,e6,e8=−e4,e7,e8=−e5
DGsetupLD
Lie algebra: alg
We also need a vector space on which we can define the adjoint representation (See Adjoint and Representation).
DGsetupw1,w2,w3,w4,w5,w6,w7,w8,W
frame name: W
ρ≔Representationalg,W,Adjointalg:
DGsetupalg,ρ,algW
Lie algebra with coefficients: algW
The linear deformations are given in terms of the Lie algebra cohomology of 𝔤 with coefficients in the adjoint representation. This cohomology is computed to be:
H2≔evalDGw2θ3&wθ6,w2θ1&wθ6−2w1θ3&wθ6+w2θ3&wθ7,w3θ1&wθ6−2w1θ1&wθ7−2w1θ2&wθ6+w2θ2&wθ7+w3θ3&wθ7,w3θ1&wθ7+w3θ2&wθ6−2w1θ2&wθ7,w3θ2&wθ7,−2w1θ1&wθ6+w2θ1&wθ7+w2θ2&wθ6+w3θ3&wθ6−2w1θ3&wθ7,w8θ6&wθ7
H2:=w2θ3⋀θ6,w2θ1⋀θ6−2w1θ3⋀θ6+w2θ3⋀θ7,w3θ1⋀θ6−2w1θ1⋀θ7−2w1θ2⋀θ6+w2θ2⋀θ7+w3θ3⋀θ7,w3θ1⋀θ7+w3θ2⋀θ6−2w1θ2⋀θ7,w3θ2⋀θ7,−2w1θ1⋀θ6+w2θ1⋀θ7+w2θ2⋀θ6+w3θ3⋀θ6−2w1θ3⋀θ7,w8θ6⋀θ7
We note that the 2-forms in H2 are all closed.
ExteriorDerivativeH2
0θ1⋀θ2⋀θ3,0θ1⋀θ2⋀θ3,0θ1⋀θ2⋀θ3,0θ1⋀θ2⋀θ3,0θ1⋀θ2⋀θ3,0θ1⋀θ2⋀θ3,0θ1⋀θ2⋀θ3
Example 1.
We consider the Lie algebra deformation defined by the first cohomology class, represented by H21.
η1≔H21
η1:=w2θ3⋀θ6
LD1≔Deformationalg,η1,κ,algD1
LD1:=e3,e6=−κe2,e4,e6=−e3,e4,e7=−e1,e5,e6=−e1,e5,e7=−e2,e6,e8=−e4,e7,e8=−e5
DGsetupLD1
Lie algebra: algD1
We use the Query command to check that this deformation defines a Lie algebra.
QueryJacobi
true
Example 2.
Here we look at the Lie algebra deformation defined by the third cohomology class, represented by H23.
η1≔H23
η1:=w3θ1⋀θ6−2w1θ1⋀θ7−2w1θ2⋀θ6+w2θ2⋀θ7+w3θ3⋀θ7
LD2≔Deformationalg,η1,κ,algD2
LD2:=e1,e6=−κe3,e1,e7=2κe1,e2,e6=2κe1,e2,e7=−κe2,e3,e7=−κe3,e4,e6=−e3,e4,e7=−e1,e5,e6=−e1,e5,e7=−e2,e6,e8=−e4,e7,e8=−e5
DGsetupLD2
Lie algebra: algD2
This time the linear deformation defined by η1 does not satisfied the Jacobi identity.
false
To continue, we calculate the quadratic deformation. For this, we need the Massey product of η1with itself.
ζ2≔MasseyProductη1,η1
ζ2:=3w3θ1⋀θ6⋀θ7+6w1θ2⋀θ6⋀θ7
Next we use the command CohomologyDecomposition to determine if the Massey product ζ2=η1, η1 is exact.
C,η2≔CohomologyDecomposition−ζ2,
C,η2:=0θ1⋀θ2⋀θ3,3w4θ1⋀θ7+6w5θ2⋀θ7−3w8θ5⋀θ7
The 3-form ζ2 is exact. The second order deformation term is given by dη2 + ζ2 =0.
ExteriorDerivativeη2&plusζ2
0θ1⋀θ2⋀θ3
We find the second order deformation to the original Lie algebra.
LD22≔Deformationalg,η1,η2,κ,algD22
LD22:=e1,e6=−κe3,e1,e7=2κe1−3κ2e4,e2,e6=2κe1,e2,e7=−κe2−6κ2e5,e3,e7=−κe3,e4,e6=−e3,e4,e7=−e1,e5,e6=−e1,e5,e7=−e2+3κ2e8,e6,e8=−e4,e7,e8=−e5
DGsetupLD22
Lie algebra: algD22
The second order deformation also fails to satisfy the Jacobi identity so we repeat the previous steps to find the third order deformation.
ζ3≔MasseyProductη1,η2&plusMasseyProductη2,η1
ζ3:=−6w4θ2⋀θ6⋀θ7
C,η3≔CohomologyDecomposition−ζ3,
C,η3:=0θ1⋀θ2⋀θ3,6w8θ2⋀θ7
The next Massey products are zero. This means that the third order deformation is a Lie algebra.
MasseyProductη1,η3
MasseyProductη2,η2
LD23≔Deformationalg,η1,η2,η3,κ,algD23
LD23:=e1,e6=−κe3,e1,e7=2κe1−3κ2e4,e2,e6=2κe1,e2,e7=−κe2−6κ2e5−6κ3e8,e3,e7=−κe3,e4,e6=−e3,e4,e7=−e1,e5,e6=−e1,e5,e7=−e2+3κ2e8,e6,e8=−e4,e7,e8=−e5
DGsetupLD23
Lie algebra: algD23
Example 3.
Here we using the calling sequence with the keyword argument parameters to find the most general linear deformation that can be constructed from the first 4 cohomology classes in H2.
η1≔evalDGa1H21+a2H22+a3H23+a4H24
η1:=a3w3+a2w2θ1⋀θ6−−a4w3+2a3w1θ1⋀θ7−−a4w3+2a3w1θ2⋀θ6+−2a4w1+a3w2θ2⋀θ7+−2a2w1+a1w2θ3⋀θ6+a3w3+a2w2θ3⋀θ7
TF,JacobiEq,JacobiSoln,LD3≔Deformationalg,η1,κ,algD3,parameters=a1,a2,a3,a4
TF,JacobiEq,JacobiSoln,LD3:=true,0,3κ2a32,6κ2a32,−3κ2a2a4,3κ2a2a4,−κ2a1a4+4κ2a2a3,8κ2a2a3−2κ2a1a4,a1=a1,a2=a2,a3=0,a4=0,a1=0,a2=0,a3=0,a4=a4,e1,e6=−κa2e2,e3,e6=2κa2e1−κa1e2,e3,e7=−κa2e2,e4,e6=−e3,e4,e7=−e1,e5,e6=−e1,e5,e7=−e2,e6,e8=−e4,e7,e8=−e5,e1,e7=−κa4e3,e2,e6=−κa4e3,e2,e7=2κa4e1,e4,e6=−e3,e4,e7=−e1,e5,e6=−e1,e5,e7=−e2,e6,e8=−e4,e7,e8=−e5
We therefore have two possibilities.The first is
DGsetupLD31
Lie algebra: algD3_1
MultiplicationTableLieTable
and the second is
DGsetupLD32
Lie algebra: algD3_2
See Also
DifferentialGeometry
LieAlgebras
Cohomology
KostantCodifferential
KostantLaplacian
MasseyProduct
Query
Representation
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