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LieAlgebras[ChangeBasis] - change the basis for a representation, either in the Lie algebra or in the representation space
Calling Sequences
ChangeBasis(rho, B, Fr)
ChangeBasis(rho, P, keyword, Fr)
Parameters
rho - a representation of a Lie algebra g on a vector space V
B - a list of vectors defining a new basis for either g or V
Fr - a Maple name or string, the name of the Lie algebra or vector space with the new basis B
P - a change of basis matrix, the columns of which are the components of the new basis vectors B with respect to the original basis
keyword - either "Domain" or "Range", indicating that the matrix A is a change of basis matrix for the Lie algebra or the representation space
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Description
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Let rho: g -> gl(V) be a representation of a Lie algebra g on a vector space V. Let [e_i] be the given basis for g and let [E_r] be the given basis for V. Let rho(e_i) = M_i, the matrix representing the linear transformation rho(e_i) with respect to the basis [E_r].
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If B = [F_r] is another basis for V, then ChangeBasis(rho, B, Fr) computes the matrices N_i for the representation rho given with respect to the basis [ F_r]. If P is the change of basis matrix whose columns are the components of the [F_r] with respect to the basis [E_r], then N_i = P^(- 1) M_i P. The command ChangeBasis(rho, P, "Range", Fr) produces the same result.
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If B = [f_i] is another basis for g, then phi = ChangeBasis(rho, B, Fr) computes the matrices N_i for the representation rho with respect to the basis [f_i]. For example, if f_1 = c1 e1 + c2 e2 + ..., then phi(f1) = c1 M_1 + c2 M_2 + ... . If P is the change of basis matrix whose columns are the components of the [f_i] with respect to the basis [e_i], then the command ChangeBasis(rho, P, "Domain", Fr) produces the same result.
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Examples
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Example 1.
We define a representation and make a change of basis for the representation space.
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![L := LieAlgebraData([[x1, x3] = x3, [x1, x4] = x4, [x2, x4] = x3], [x1, x2, x3, x4], Alg1)](/support/helpjp/helpview.aspx?si=6626/file05792/math157.png)
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![_DG([["LieAlgebra", Alg1, [4, table( [ ] )]], [[[1, 3, 3], 1], [[1, 4, 4], 1], [[2, 4, 3], 1]]])](/support/helpjp/helpview.aspx?si=6626/file05792/math160.png)
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Alg1 >
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![DGsetup([x, y, z], V)](/support/helpjp/helpview.aspx?si=6626/file05792/math171.png)
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| (2.3) |
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![M := [Matrix([[1, 0, 0], [0, 1, 0], [0, 0, 0]]), Matrix([[0, 1, 0], [0, 0, 0], [0, 0, 0]]), Matrix([[0, 0, 1], [0, 0, 0], [0, 0, 0]]), Matrix([[0, 0, 0], [0, 0, 1], [0, 0, 0]])]](/support/helpjp/helpview.aspx?si=6626/file05792/math178.png)
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Define the new basis for the representation space.
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Alg1 >
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![B := evalDG([D_x+D_y+D_z, D_x-D_y, D_x+2*D_y+D_z])](/support/helpjp/helpview.aspx?si=6626/file05792/math193.png)
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![[_DG([["vector", V, []], [[[1], 1], [[2], 1], [[3], 1]]]), _DG([["vector", V, []], [[[1], 1], [[2], -1]]]), _DG([["vector", V, []], [[[1], 1], [[2], 2], [[3], 1]]])]](/support/helpjp/helpview.aspx?si=6626/file05792/math196.png)
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Compute the representation rho in the basis Fr.
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We can use the Query command to check that phi1 is a representation of Alg1.
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Alg1 >
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Check, by example, that the matrices for phi1 are correct. We apply rho(e1) to Fr[1] and express the result as a linear combination of the vectors Fr. This should give the first column of the matrix for e1 in phi1.
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Alg1 >
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![a := ApplyRepresentation(rho, e1, B[1])](/support/helpjp/helpview.aspx?si=6626/file05792/math249.png)
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![_DG([["vector", V, []], [[[1], 1], [[2], 1]]])](/support/helpjp/helpview.aspx?si=6626/file05792/math252.png)
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![[-2, 1, 2]](/support/helpjp/helpview.aspx?si=6626/file05792/math259.png)
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Example 2.
We obtain the same change of basis as in Example 1 using the other calling sequence for the procedure ChangeBasis. We take the matrix A to be the matrix whose columns are the coefficients of the new basis in terms of the old.
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![[_DG([["vector", V, []], [[[1], 1], [[2], 1], [[3], 1]]]), _DG([["vector", V, []], [[[1], 1], [[2], -1]]]), _DG([["vector", V, []], [[[1], 1], [[2], 2], [[3], 1]]])]](/support/helpjp/helpview.aspx?si=6626/file05792/math276.png)
| (2.10) |
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![P := LinearAlgebra:-Transpose(Matrix(GetComponents(B, [D_x, D_y, D_z])))](/support/helpjp/helpview.aspx?si=6626/file05792/math280.png)
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Example 3.
Now we make a change of basis in the Lie algebra. First we use the LieAlgebraData command to create the Lie algebra in the new basis.
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Alg1 >
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![B := evalDG([e1+e2, e3-e2, e2-2*e3, e1-e3+e4])](/support/helpjp/helpview.aspx?si=6626/file05792/math311.png)
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![[_DG([["vector", Alg1, []], [[[1], 1], [[2], 1]]]), _DG([["vector", Alg1, []], [[[2], -1], [[3], 1]]]), _DG([["vector", Alg1, []], [[[2], 1], [[3], -2]]]), _DG([["vector", Alg1, []], [[[1], 1], [[3], -1], [[4], 1]]])]](/support/helpjp/helpview.aspx?si=6626/file05792/math314.png)
| (2.14) |
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Alg1 >
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![_DG([["LieAlgebra", Alg2, [4, table( [ ] )]], [[[1, 2, 2], -1], [[1, 2, 3], -1], [[1, 3, 2], 2], [[1, 3, 3], 2], [[1, 4, 1], -1], [[1, 4, 2], -3], [[1, 4, 3], -2], [[1, 4, 4], 1], [[2, 4, 2], 2], [[2, 4, 3], 2], [[3, 4, 2], -3], [[3, 4, 3], -3]]])](/support/helpjp/helpview.aspx?si=6626/file05792/math321.png)
| (2.15) |
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Alg1 >
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![DGsetup(L2, [f], [theta])](/support/helpjp/helpview.aspx?si=6626/file05792/math325.png)
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| (2.16) |
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Alg2 >
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| (2.17) |
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Alg2 >
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Example 4.
We obtain the same change of basis as in Example 3 using the other calling sequence for the procedure ChangeBasis. We take the matrix A to be the matrix whose columns are the coefficients of the new basis in terms of the old.
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Alg2 >
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![[_DG([["vector", Alg1, []], [[[1], 1], [[2], 1]]]), _DG([["vector", Alg1, []], [[[2], -1], [[3], 1]]]), _DG([["vector", Alg1, []], [[[2], 1], [[3], -2]]]), _DG([["vector", Alg1, []], [[[1], 1], [[3], -1], [[4], 1]]])]](/support/helpjp/helpview.aspx?si=6626/file05792/math360.png)
| (2.19) |
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Alg1 >
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![P := LinearAlgebra:-Transpose(Matrix(GetComponents(B, [e1, e2, e3, e4])))](/support/helpjp/helpview.aspx?si=6626/file05792/math364.png)
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| (2.20) |
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Alg1 >
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| (2.21) |
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