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Query[Homomorphism] - check if a Matrix defines a Lie algebra homomorphism between two Lie algebras
Calling Sequences
Query(Alg1, Alg2, A, "Homomorphism")
Query(Alg1, Alg2, A, parm, "Homomorphism")
Parameters
Alg1 - the name of an initialized Lie algebra g, the domain algebra for the homomorphism defined by A
Alg2 - the name of an initialized Lie algebra k, the range algebra for the homomorphism defined by A
A - an m x n Matrix, where n is the dimension of the Lie algebra g and m is the dimension of k, or a transformation from Alg1 to Alg2
parm - a set of parameters appearing in the Matrix A or in the Lie algebras g and k
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Description
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A matrix A defines a Lie algebra homomorphism from g to k if the linear transformation T: g -> k determined by A satisfies T([x,y]) = [T(x), T(y)] for all x, y in g.
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Query(Alg1, Alg2, A, "Homomorphism") returns true if the matrix A defines a Lie algebra homomorphism from g to k and false otherwise.
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Query(Alg1, Alg2, parm, "Homomorphism") returns a 4-tuple TF, Eq, Soln, B. Here TF is true if Maple finds a set of values for the parameters for which the Matrix A is a homomorphism; Eq is the defining set of equations for the parameters parm in order that the matrix A be a homomorphism; Soln is a list of solutions to the equations Eq; and B is the list of Matrices obtained by evaluating A on the solutions in the list Soln.
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The command Query is part of the DifferentialGeometry:-LieAlgebras package. It can be used in the form Query(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-Query(...).
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Examples
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Example 1.
First initialize a Lie algebra. We illustrate the fact that Ad(x), for any x in the Lie algebra, is always a Lie algebra homomorphism (isomorphism).
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![L1 := _DG([["LieAlgebra", Alg1, [4]], [[[1, 4, 1], 1], [[2, 3, 1], 1], [[2, 4, 2], 1]]])](/support/helpjp/helpview.aspx?si=7158/file07651/math158.png)
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![L1 := [[e1, e4] = e1, [e2, e3] = e1, [e2, e4] = e2]](/support/helpjp/helpview.aspx?si=7158/file07651/math161.png)
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Alg1 >
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![A1 := Matrix([[1, -t, s, -t*s/2+r], [0, 1, 0, s], [0, 0, 1, 0], [0, 0, 0, 1]])](/support/helpjp/helpview.aspx?si=7158/file07651/math172.png)
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Alg1 >
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Example 2.
The Matrix exponential of any Outer derivation is also a Lie algebra homomorphism (isomorphism).
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Alg1 >
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![Outer := [Matrix([[1, 0, 0, 0], [0, -1, 0, 0], [0, 0, 2, 0], [0, 0, 0, 0]])]](/support/helpjp/helpview.aspx?si=7158/file07651/math192.png)
| (2.4) |
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Alg1 >
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![A2 := LinearAlgebra:-MatrixExponential(t*Outer[1])](/support/helpjp/helpview.aspx?si=7158/file07651/math196.png)
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![A2 := Matrix([[exp(t), 0, 0, 0], [0, exp(-t), 0, 0], [0, 0, exp(2*t), 0], [0, 0, 0, 1]])](/support/helpjp/helpview.aspx?si=7158/file07651/math199.png)
| (2.5) |
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Alg1 >
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![T := [[e1, exp(t)*e1], [e2, exp(-t)*e2], [e3, exp(2*t)*e3], [e4, e4]]](/support/helpjp/helpview.aspx?si=7158/file07651/math206.png)
| (2.6) |
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Alg1 >
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| (2.7) |
Example 3.
In this example we construct the quotient algebra of Alg1 by the ideal [e1]. Call the quotient Alg2. We check that the canonical projection map from Alg1 to Alg2 is a Lie algebra homomorphism.
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Alg1 >
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![L2 := QuotientAlgebra([e1], [e2, e3, e4], Alg2)](/support/helpjp/helpview.aspx?si=7158/file07651/math233.png)
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![L2 := [[e1, e3] = e1]](/support/helpjp/helpview.aspx?si=7158/file07651/math236.png)
| (2.8) |
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Alg1 >
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![DGsetup(L2, [x], [alpha])](/support/helpjp/helpview.aspx?si=7158/file07651/math240.png)
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| (2.9) |
The following Matrix A3 maps e1 -> 0, e2 -> x1, e3 -> x2, e4 -> x3.
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Alg2 >
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![A3 := Matrix([[0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]])](/support/helpjp/helpview.aspx?si=7158/file07651/math262.png)
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![A3 := Matrix([[0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]])](/support/helpjp/helpview.aspx?si=7158/file07651/math265.png)
| (2.10) |
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Alg2 >
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| (2.11) |
Example 4.
In this example we shall find all the monomorphisms from the 2 dimensional solvable Lie algebra into Alg1.
This effectively computes all the 2 dimensional non-Abelian subalgebras of Alg1.
First initialize the 2 dimensional solvable algebra and call it Alg3.
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Alg2 >
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![L2 := _DG([["LieAlgebra", Alg3, [2]], [[[1, 2, 1], 1]]])](/support/helpjp/helpview.aspx?si=7158/file07651/math292.png)
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![L2 := [[e1, e2] = e1]](/support/helpjp/helpview.aspx?si=7158/file07651/math295.png)
| (2.12) |
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Alg2 >
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Define a matrix A4 representing an arbitrary linear transformation from Alg1 to Alg2.
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Alg3 >
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![A4 := Matrix([[a1, a2], [a3, a4], [a5, a6], [a7, a8]])](/support/helpjp/helpview.aspx?si=7158/file07651/math313.png)
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![A4 := Matrix([[a1, a2], [a3, a4], [a5, a6], [a7, a8]])](/support/helpjp/helpview.aspx?si=7158/file07651/math316.png)
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Determine the parameter values for which A4 is a Lie algebra homomorphism.
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Alg3 >
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The equations that must hold for A4 to define a Lie algebra homomorphism are given by EQ.
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Alg1 >
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| (2.15) |
The possible Lie algebra homomorphisms are given by B. Note that B[2], B[3] and B[4] can be chosen to be full rank and therefore define Lie algebra isomorphisms.
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Alg1 >
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![[[[e1, `0`*e1], [e2, a2*e1+a4*e2+a6*e3+a8*e4]], [[e1, a1*e1], [e2, a2*e1+a4*e2+a6*e3+e4]], [[e1, a1*e1], [e2, a2*e1+a4*e2+a6*e3+e4]], [[e1, a1*e1], [e2, a2*e1+a4*e2+a6*e3+e4]], [[e1, a1*e1+a3*e2], [e2, a2*e1+a4*e2+e4]]]](/support/helpjp/helpview.aspx?si=7158/file07651/math363.png)
| (2.16) |
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Alg1 >
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