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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
Description
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].
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.
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.
Examples
Example 1.
We define a representation and make a change of basis for the representation space.
Define the new basis for the representation space.
Compute the representation rho in the basis Fr.
We can use the Query command to check that phi1 is a representation of Alg1.
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.
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.
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.
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.
See Also
DifferentialGeometry, LieAlgebras, ChangeFrame, GetComponents, LieAlgebraData, Query
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