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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

Description

• 

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.

Examples

withDifferentialGeometry:withLieAlgebras:

 

First initialize an 8-dimensional Lie algebra. We shall create various deformations of this Lie algebra. Here are the structure equations.

StrEqx4,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

(2.1)

 

Use the commands LieAlgebraData and DGsetup to initialize this Lie algebra.

LDLieAlgebraDataStrEq,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

(2.2)

DGsetupLD

Lie algebra: alg

(2.3)

 

We also need a vector space on which we can define the adjoint representation (See Adjoint and Representation).

alg > 

DGsetupw1,w2,w3,w4,w5,w6,w7,w8,W

frame name: W

(2.4)
alg > 

ρRepresentationalg,W,Adjointalg:

alg > 

DGsetupalg,ρ,algW

Lie algebra with coefficients: algW

(2.5)

 

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:

H2evalDGw2θ3 &w θ6,w2θ1 &w θ62w1θ3 &w θ6+w2θ3 &w θ7,w3θ1 &w θ62w1θ1 &w θ72w1θ2 &w θ6+w2θ2 &w θ7+w3θ3 &w θ7,w3θ1 &w θ7+w3θ2 &w θ62w1θ2 &w θ7,w3θ2 &w θ7,2w1θ1 &w θ6+w2θ1 &w θ7+w2θ2 &w θ6+w3θ3 &w θ62w1θ3 &w θ7,w8θ6 &w θ7

H2:=w2θ3θ6,w2θ1θ62w1θ3θ6+w2θ3θ7,w3θ1θ62w1θ1θ72w1θ2θ6+w2θ2θ7+w3θ3θ7,w3θ1θ7+w3θ2θ62w1θ2θ7,w3θ2θ7,2w1θ1θ6+w2θ1θ7+w2θ2θ6+w3θ3θ62w1θ3θ7,w8θ6θ7

(2.6)

 

 We note that the 2-forms in H2 are all closed.

ExteriorDerivativeH2

0θ1θ2θ3,0θ1θ2θ3,0θ1θ2θ3,0θ1θ2θ3,0θ1θ2θ3,0θ1θ2θ3,0θ1θ2θ3

(2.7)

 

Example 1.

We consider the Lie algebra deformation defined by the first cohomology class, represented by H21.

 

algD1 > 

η1H21

η1:=w2θ3θ6

(2.8)
algW > 

LD1Deformationalg,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

(2.9)
alg > 

DGsetupLD1

Lie algebra: algD1

(2.10)

 

We use the Query command to check that this deformation defines a Lie algebra.

algD1 > 

QueryJacobi

true

(2.11)

 

Example 2.

Here we look at the Lie algebra deformation defined by the third cohomology class, represented by H23.

 

algD1 > 

η1H23

η1:=w3θ1θ62w1θ1θ72w1θ2θ6+w2θ2θ7+w3θ3θ7

(2.12)
algW > 

LD2Deformationalg,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

(2.13)
alg > 

DGsetupLD2

Lie algebra: algD2

(2.14)

 

This time the linear deformation defined by η1 does not satisfied the Jacobi identity.

algD2 > 

QueryJacobi

false

(2.15)

 

To continue, we calculate the quadratic deformation. For this, we need the Massey product of η1with itself.

algW > 

ζ2MasseyProductη1,η1

ζ2:=3w3θ1θ6θ7+6w1θ2θ6θ7

(2.16)

 

Next we use the command CohomologyDecomposition to determine if the Massey product  ζ2=η1, η1 is exact.

algW > 

C,η2CohomologyDecompositionζ2,

C,η2:=0θ1θ2θ3,3w4θ1θ7+6w5θ2θ73w8θ5θ7

(2.17)

 

The 3-form ζ2 is exact. The second order deformation term is given by dη2 + ζ2 =0.

algW > 

ExteriorDerivativeη2 &plus ζ2

0θ1θ2θ3

(2.18)

 

We find the second order deformation to the original Lie algebra.

Alg1 > 

LD22Deformationalg,η1,η2,κ,algD22

LD22:=e1,e6=κe3,e1,e7=2κe13κ2e4,e2,e6=2κe1,e2,e7=κe26κ2e5,e3,e7=κe3,e4,e6=e3,e4,e7=e1,e5,e6=e1,e5,e7=e2+3κ2e8,e6,e8=e4,e7,e8=e5

(2.19)
alg > 

DGsetupLD22

Lie algebra: algD22

(2.20)
alg > 

QueryJacobi

false

(2.21)

 

The second order deformation also fails to satisfy the Jacobi identity so we repeat the previous steps to find the third order deformation.

algD22 > 

ζ3MasseyProductη1,η2 &plus MasseyProductη2,η1

ζ3:=6w4θ2θ6θ7

(2.22)
algD22 > 

C,η3CohomologyDecompositionζ3,

C,η3:=0θ1θ2θ3,6w8θ2θ7

(2.23)

 

The next Massey products are zero. This means that the third order deformation is a Lie algebra.

algW > 

MasseyProductη1,η3

0θ1θ2θ3

(2.24)
algW > 

MasseyProductη2,η2

0θ1θ2θ3

(2.25)
Alg1 > 

LD23Deformationalg,η1,η2,η3,κ,algD23

LD23:=e1,e6=κe3,e1,e7=2κe13κ2e4,e2,e6=2κe1,e2,e7=κe26κ2e56κ3e8,e3,e7=κe3,e4,e6=e3,e4,e7=e1,e5,e6=e1,e5,e7=e2+3κ2e8,e6,e8=e4,e7,e8=e5

(2.26)
alg > 

DGsetupLD23

Lie algebra: algD23

(2.27)
algW > 

QueryJacobi

true

(2.28)

 

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.

 

algD23 > 

η1evalDGa1H21+a2H22+a3H23+a4H24

η1:=a3w3+a2w2θ1θ6a4w3+2a3w1θ1θ7a4w3+2a3w1θ2θ6+2a4w1+a3w2θ2θ7+2a2w1+a1w2θ3θ6+a3w3+a2w2θ3θ7

(2.29)
algD24 > 

TF,JacobiEq,JacobiSoln,LD3Deformationalg,η1,κ,algD3,parameters=a1,a2,a3,a4

TF,JacobiEq,JacobiSoln,LD3:=true,0,3κ2a32,6κ2a32,3κ2a2a4,3κ2a2a4,κ2a1a4+4κ2a2a3,8κ2a2a32κ2a1a4,a1=a1,a2=a2,a3=0,a4=0,a1=0,a2=0,a3=0,a4=a4,e1,e6=κa2e2,e3,e6=2κa2e1κa1e2,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

(2.30)

 

We therefore have two possibilities.The first is

algD24 > 

DGsetupLD31

Lie algebra: algD3_1

(2.31)
algD24 > 

MultiplicationTableLieTable

| e1e2e3e4e5e6e7e8------------------------------------e1| 00000κa2e200e2| 00000000e3| 000002κa2e1κa1e2κa2e20e4| 00000e3e10e5| 00000e1e20e6| κa2e202κa2e1+κa1e2e3e100e4e7| 00κa2e2e1e200e5e8| 00000e4e50

(2.32)

 

and the second is

algD24 > 

DGsetupLD32

Lie algebra: algD3_2

(2.33)
algD24 > 

MultiplicationTableLieTable

| e1e2e3e4e5e6e7e8------------------------------------e1| 000000κa4e30e2| 00000κa4e32κa4e10e3| 00000000e4| 00000e3e10e5| 00000e1e20e6| 0κa4e30e3e100e4e7| κa4e32κa4e10e1e200e5e8| 00000e4e50

(2.34)

See Also

DifferentialGeometry

LieAlgebras

Cohomology

KostantCodifferential

KostantLaplacian

MasseyProduct

Query

Representation

 


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