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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
Description
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.
Query(Alg1, Alg2, A, "Homomorphism") returns true if the matrix A defines a Lie algebra homomorphism from g to k and false otherwise.
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.
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(...).
Examples
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).
Example 2.
The Matrix exponential of any Outer derivation is also a Lie algebra homomorphism (isomorphism).
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.
The following Matrix A3 maps e1 -> 0, e2 -> x1, e3 -> x2, e4 -> x3.
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.
Define a matrix A4 representing an arbitrary linear transformation from Alg1 to Alg2.
Determine the parameter values for which A4 is a Lie algebra homomorphism.
The equations that must hold for A4 to define a Lie algebra homomorphism are given by EQ.
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.
See Also
DifferentialGeometry, LieAlgebras, Adjoint, Derivations, Query, QuotientAlgebra, Transformation
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