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LieAlgebras[MasseyProduct] - calculate the Massey product of a pair of forms

Calling Sequences

     MasseyProduct(α,β)

Parameters

      α    - a p-form defined on a Lie algebra 𝔤 with coefficients in 𝔤

      β     - a q-form defined on a Lie algebra 𝔤 with coefficients in 𝔤

 

Description

Examples

Description

• 

The Massey product of a pair of forms α  Λ2𝔤, 𝔤 and β  Λ2𝔤, 𝔤 is the 3-form α, β defined by

α, βx, y, z= αβx, y, z + αβz, x, y + αβy, z, x.

In general, if α ϵ Λp𝔤, 𝔤 and β ϵ Λq𝔤, 𝔤,then the Massey product is the p+q 1form defined by

 

α,βx1, ... ,xp+q1 = αβx1,...,xq,xq+1 ,... ,xq+p1 + cyclic permutations.

 

• 

 The Massey product plays an important role in the construction of the deformations of a Lie algebra.

 

Examples

withDifferentialGeometry:withLieAlgebras:

 

Example 1.

First initialize a Lie algebra from a list of structure equations.

 

StrEqx2,x3=x1,x2,x5=x3,x4,x5=x4

StrEq:=x2,x3=x1,x2,x5=x3,x4,x5=x4

(2.1)

LDLieAlgebraDataStrEq,x1,x2,x3,x4,x5,alg

LD:=e2,e3=e1,e2,e5=e3,e4,e5=e4

(2.2)

DGsetupLD

Lie algebra: alg

(2.3)

 

We define the adjoint representation and use this to construct the corresponding Lie algebra with coefficients.

alg > 

DGsetupw1,w2,w3,w4,w5,V

frame name: V

(2.4)
alg > 

ρRepresentationalg,V,Adjointalg:

alg > 

DGsetupalg,ρ,algV

Lie algebra with coefficients: algV

(2.5)

 

Here is a pair of 2-forms on algV and their Massey product.

alg > 

αevalDGw1θ1&wθ2

α:=w1θ1θ2

(2.6)
algV > 

βevalDGw2θ1&wθ4

β:=w2θ1θ4

(2.7)
algV > 

MasseyProductα,β

w2θ1θ2θ4

(2.8)

 

Here is a pair of 3-forms on algV and their Massey product.

algV > 

αevalDGw1θ1&wθ2&wθ3

α:=w1θ1θ2θ3

(2.9)
algV > 

βevalDGw4θ1&wθ4&wθ5

β:=w4θ1θ4θ5

(2.10)
algV > 

MasseyProductα,β

w4θ1θ2θ3θ4θ5

(2.11)

See Also

DifferentialGeometry

LieAlgebras

Adjoint

Cohomology

Deformation

Representation