LieAlgebras[QuotientAlgebra] - create the Lie algebra data structure for a quotient algebra of a Lie algebra by an ideal
Calling Sequences
QuotientAlgebra(h, m, Algname, keyword)
Parameters
h - a list of independent vectors defining an ideal h in a Lie algebra g
m - a list of independent vectors defining a vector space complement to h in g
Algname - a name or a string, the name assigned to the quotient algebra g/h
keyword - (optional) the keyword "Matrix"
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Description
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Let g be a Lie algebra and h an ideal in g. Then elements of the quotient algebra g/h are the h cosets x + h, where x in g. The Lie bracket on g/h is defined by [x + h, y + h] = [x, y] + h. If vectors y_1, y_2, .. y_r form a basis for a complement to h, then the cosets y_1 + h, y_2 + h, ..., y_r + h form a basis for g/h.
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The program QuotientAlgebra(h, m) creates a Lie algebra data structure for the quotient algebra g/h using the vectors in the complement m as the representative basis elements for g/h.
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A Lie algebra data structure contains the structure constants in a standard format used by the LieAlgebras package. 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.
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With the optional keyword present, QuotientAlgebra(h, m, "Matrix") returns the Lie algebra data structure for g/h and the Matrix representation of the canonical projection map g -> g/h defined by x -> x + h.
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The command QuotientAlgebra is part of the DifferentialGeometry:-LieAlgebras package. It can be used in the form QuotientAlgebra(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-QuotientAlgebra(...).
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Examples
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Example 1.
First initialize a Lie algebra and display the multiplication table.
Alg2 >
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| (2.1) |
Alg2 >
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Check that S1 = [e1, e3] is an ideal and find the quotient algebra (call it Alg2) using the complement vectors [e2, e4, e5].
Alg1 >
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Alg1 >
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| (2.2) |
Alg1 >
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Rerun QuotientAlgebra with the keyword argument "Matrix".
Alg1 >
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Alg2 >
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We use the DifferentialGeometry command transform to convert the matrix A into a transformation from Alg1 to the quotient algebra Alg2.
Alg1 >
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| (2.5) |
We can check that PI is a Lie algebra homomorphism.
Alg2 >
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| (2.6) |
We see that PI sends e1 to 0, f2 to f1 and so on.
Alg2 >
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| (2.7) |
We can verify that [e1, e3] is the kernel of PI and that the image of PI is [f1, f2, f3] (so that PI is surjective).
Alg2 >
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| (2.8) |
Alg1 >
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| (2.9) |
Alg2 >
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