LieAlgebras[Extension] - calculate a right or a central extension of a Lie algebra
Extension(AlgName1, A, AlgName2)
Extension(AlgName1, β, AlgName2)
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
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(...).
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 ≔ _DG⁡LieAlgebra,Alg1,4,1,4,1,2,2,3,1,1,2,4,2,1,3,4,2,1,3,4,3,1
Here are two derivations we shall use to make right extensions.
A1 ≔ Adjoint⁡e1−e2+2⁢e3−e4;A2 ≔ Derivations⁡Outer1
Use the matrix A1 to make a right extension.
L2 ≔ Extension⁡Alg1,A1,Alg2
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.
Lie algebra: Alg2
Repeat these computations using the outer derivation A2.
L3 ≔ Extension⁡Alg1,A2,Alg3
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.
Lie algebra: Alg3
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.
L4 ≔ _DG⁡LieAlgebra,Alg4,4,2,4,1,1,3,4,3,1
Define a pair of 2-forms and check that they are closed.
β1 ≔ θ2 &wedge θ4;β2 ≔ θ1 &wedge θ4
Use β1 to make a central extension.
L5 ≔ Extension⁡Alg4,β1,Alg5
Initialize this Lie algebra. Since the form β1 is exact, this central extension is trivial.
Now make the central extension using β2. This extension is indecomposable.
L6 ≔ Extension⁡Alg4,β2,Alg6
DifferentialGeometry, LieAlgebras, Adjoint, Decompose, Derivations, ExteriorDerivative, MultiplicationTable, Query
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