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LieAlgebraData[MatrixAlgebra] - convert a list of Matrices to a Lie algebra data structure
Calling Sequence
LieAlgebraConvert(MatrixAlgebra, AlgName)
Parameters
MatrixAlgebra - a list of linearly independent square Matrices (all of the same dimension)
AlgName - a name or string, the name to be assigned to the Lie algebra
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Description
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In the LieAlgebras package, the command DGsetup is used to initialize a Lie algebra -- that is, to define the basis elements for the Lie algebra and its dual and to store the structure constants for the Lie algebra in memory. The first argument for DGsetup is a Lie algebra data structure which contains the structure constants in a standard format used by the LieAlgebras package.
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An important class of Lie algebras is given by the Matrix Lie algebras, where the Lie algebra multiplication is given by the matrix commutator [a, b] = a.b - b.a. The program LieAlgebraData(MatrixAlgebra, AlgName) will create the Lie algebra data structure for the matrix algebra whose basis is the given list of matrices MatrixAlgebra.
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The command LieAlgebraData is part of the DifferentialGeometry:-LieAlgebras package. It can be used in the form LieAlgebraData(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-LieAlgebraData(...).
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Examples
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Example 1.
In this example we create a 5 dimensional Lie algebra from a list of 3 x 3 matrices.
First we define the matrices.
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![M := map(Matrix, [[[-1, 2, -1], [0, 1, 0], [2, -2, 2]], [[1, 0, 0], [2, -1, 1], [2, -2, 2]], [[0, 0, 0], [1, -1, 1], [2, -2, 2]], [[-1, 2, -1], [1, 0, 0], [4, -4, 2]], [[4, -4, 2], [2, -2, 1], [-2, 2, -1]]])](/support/helpjp/helpview.aspx?si=7153/file07637/math68.png)
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![M := [Matrix([[-1, 2, -1], [0, 1, 0], [2, -2, 2]]), Matrix([[1, 0, 0], [2, -1, 1], [2, -2, 2]]), Matrix([[0, 0, 0], [1, -1, 1], [2, -2, 2]]), Matrix([[-1, 2, -1], [1, 0, 0], [4, -4, 2]]), Matrix([[4, -4, 2], [2, -2, 1], [-2, 2, -1]])]](/support/helpjp/helpview.aspx?si=7153/file07637/math71.png)
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Before continuing we note that it is not difficult to directly check that the span of the Matrices M is closed under matrix commutators. For example, the matrix commutator of M[1] and M[5] is - M[1] + M[2]. (The matrices P and Q computed below are equal.)
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![P := M[1].M[5]-M[5].M[1]](/support/helpjp/helpview.aspx?si=7153/file07637/math91.png)
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![P := Matrix([[2, -2, 1], [2, -2, 1], [0, 0, 0]])](/support/helpjp/helpview.aspx?si=7153/file07637/math94.png)
| (2.2) |
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![[-1, 1, 0, 0, 0]](/support/helpjp/helpview.aspx?si=7153/file07637/math101.png)
| (2.3) |
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![Q := -M[1]+M[2]](/support/helpjp/helpview.aspx?si=7153/file07637/math105.png)
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![Q := Matrix([[2, -2, 1], [2, -2, 1], [0, 0, 0]])](/support/helpjp/helpview.aspx?si=7153/file07637/math108.png)
| (2.4) |
The program LieAlgebraData will calculate all the required matrix commutators of the matrices in M as linear combinations in M and return the result as a Lie algebra data structure.
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![L := [[e1, e2] = -e1+e2, [e1, e4] = -e1+e3+e4, [e1, e5] = -e1+e2, [e2, e4] = -e2+e3+e4, [e2, e5] = -2*e1+2*e2, [e3, e4] = -e1+e3+e4, [e4, e5] = -2*e1+e2+e3+e4]](/support/helpjp/helpview.aspx?si=7153/file07637/math125.png)
| (2.5) |
Here e1, e2, ..., e5 represent the matrices M[1], M[2], ..., M[5] respectively. Thus the structure equation [e1, e5] = - e1 + e2 coincides with that explicitly computed above.
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Finally, we remark that this Lie algebra is decomposable.
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Ex1 >
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| (2.6) |
A decomposition of this Lie algebra can be found using the program DecomposeAlgebra.
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