calculate a right or a central extension of a Lie algebra - Maple Programming Help

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LieAlgebras[Extension] - calculate a right or a central extension of a Lie algebra

Calling Sequences

     Extension(AlgName1, A, AlgName2)

     Extension(AlgName1, β, AlgName2)

Parameters

     AlgName1 - a name or string, the name of the Lie algebra 𝔤 to be extended

     A        - a transformation, defining derivation of 𝔤

     β        - a closed 2-form

     AlgName2 - a name or string, the name to be given to the Lie algebra extension

 

Description

Examples

See Also

Description

• 

Let 𝔤 be a Lie algebra and let φ : 𝔤  𝔤 be a derivation on 𝔤. Then the right extension of 𝔤 by φ is the Lie algebra k = 𝔤 + ℝ with Lie bracket

 x, a,, (y, b] = ( x, y + b φx  a φy, 0) for all x, y 𝔤 and a,b ℝ. 

The right extension k is said to be trivial if k splits as a Lie algebra direct sum k = 𝔤' ℝ, where 𝔤' is isomorphic to 𝔤. The extension k is trivial precisely when φ is an inner derivation.

• 

Let 𝔤 be a Lie algebra and let β be a closed 2-form on 𝔤. Then the central extension of 𝔤 by β is the Lie algebra k = 𝔤 + ℝ with Lie bracket

 x, a,, (y, b] = x, y, βx, y for all x, y 𝔤 and a,b ℝ. 

The extension k is said to be trivial if k splits as a Lie algebra direct sum k = 𝔤' ℝ, where 𝔤' is isomorphic to 𝔤. The extension k is trivial precisely when β is exact, that is, β = dη.

• 

Extension computes a right extension when its second argument is a matrix and a central extension when the second argument is a 2-form. The procedure returns the Lie algebra data structure for the extended algebra. The structure equations for the extension are displayed.

• 

The command Extension is part of the DifferentialGeometry:-LieAlgebras package. It can be used in the form Extension(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-Extension(...).

Examples

withDifferentialGeometry:withLieAlgebras:

 

Example 1.

Calculate two right extensions and show that the first is trivial and the second is not. First initialize the Lie algebra Alg1 and display the multiplication table.

L1_DGLieAlgebra,Alg1,4,1,4,1,2,2,3,1,1,2,4,2,1,3,4,2,1,3,4,3,1

L1:=e1,e4=2e1,e2,e3=e1,e2,e4=e2,e3,e4=e2+e3

(2.1)

DGsetupL1:

 

Here are two derivations we shall use to make right extensions.

Alg1 > 

A1Adjointe1e2+2e3e4;A2DerivationsOuter1

A1:=2212011100120000

A2:=100001200001200000

(2.2)

 

Use the matrix A1 to make a right extension.

Alg1 > 

L2ExtensionAlg1,A1,Alg2

L2:=e1,e4=2e1,e1,e5=2e1,e2,e3=e1,e2,e4=e2,e2,e5=2e1+e2,e3,e4=e2+e3,e3,e5=e1+e2+e3,e4,e5=2e1+e2+2e3

(2.3)

 

Initialize this Lie algebra. Since it was constructed using an inner derivation, it should be a trivial extension. This we check using the Decompose command.

Alg1 > 

DGsetupL2

Lie algebra: Alg2

(2.4)
Alg2 > 

QueryAlg2,Indecomposable

false

(2.5)

 

Repeat these computations using the outer derivation A2.

Alg2 > 

L3ExtensionAlg1,A2,Alg3

L3:=e1,e4=2e1,e1,e5=e1,e2,e3=e1,e2,e4=e2,e2,e5=12e2,e3,e4=e2+e3,e3,e5=12e3

(2.6)

 

Initialize this right extension. Since it was constructed using an outer derivation, it should be not be a trivial extension. This we check using the Decompose command.

Alg1 > 

DGsetupL3

Lie algebra: Alg3

(2.7)
Alg3 > 

QueryAlg3,Indecomposable

true

(2.8)

 

Example 2.

Calculate two central extensions and show that the first is trivial and the second is not. First initialize the Lie algebra Alg4 and display the multiplication table. Now display the exterior derivatives of the 1-forms for Alg1.

Alg3 > 

L4_DGLieAlgebra,Alg4,4,2,4,1,1,3,4,3,1

L4:=e2,e4=e1,e3,e4=e3

(2.9)
Alg3 > 

DGsetupL4:

Alg4 > 

MultiplicationTableAlg4,ExteriorDerivative

dθ1=θ2θ4

dθ2=0θ1θ2

dθ3=θ3θ4

dθ4=0θ1θ2

(2.10)

 

Define a pair of 2-forms and check that they are closed.

Alg4 > 

β1θ2 &wedge θ4;β2θ1 &wedge θ4

β1:=θ2θ4

β2:=θ1θ4

(2.11)
Alg4 > 

ExteriorDerivativeβ1,ExteriorDerivativeβ2

0θ1θ2θ3,0θ1θ2θ3

(2.12)

 

Use β1 to make a central extension.

Alg4 > 

L5ExtensionAlg4,β1,Alg5

L5:=e2,e4=e1+e5,e3,e4=e3

(2.13)

 

Initialize this Lie algebra. Since the form β1 is exact, this central extension is trivial.

Alg4 > 

DGsetupL5:

Alg5 > 

QueryAlg4,Indecomposable

true

(2.14)

 

Now make the central extension using β2. This extension is indecomposable.

Alg4 > 

L6ExtensionAlg4,β2,Alg6

L6:=e1,e4=e5,e2,e4=e1,e3,e4=e3

(2.15)
Alg4 > 

DGsetupL6:

Alg6 > 

QueryAlg6,Indecomposable

true

(2.16)

See Also

DifferentialGeometry, LieAlgebras, Adjoint, Decompose, Derivations, ExteriorDerivative, MultiplicationTable, Query


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