Query[NaturallyReductivePair] - check if a subalgebra, subspace pair is naturally reductive with respect to an inner product on the subspace
Calling Sequences
Query(S, M, B, "NaturallyReductivePair")
Query(S, M, B, parm, "NaturallyReductivePair")
Parameters
S - a list of independent vectors which defines a subalgebra in a Lie algebra g
M - a list of independent vectors which defines a reductive complement to S in g
B - a symmetric m x m matrix, which defines an inner product on M with respect to the given basis
parm - (optional) a set of parameters appearing in the list of vectors M
|
Description
|
|
•
|
A subalgebra, subspace pair S, M is naturally reductive with respect to an inner product B on M if [i] the subspace M defines a reductive complement to the subalgebra S, and [ii] if B is a non-degenerate g invariant inner product on M, that is B([x, y]_M, z) + B(y, [x, z]_M) = 0 for all x in g and for all y and z in M. Here [x, y]_M refers to the component of [x, y] in M with respect to the direct sum decomposition g = S + M.
|
•
|
Query(S, M, B, "NaturallyReductivePair") returns true if S, M is naturally reductive with respect to the inner product B, and false otherwise.
|
•
|
Query(S, M, B, parm, "NaturallyReductivePair") returns a sequence TF, Eq, Soln, NatRedPair. Here TF is true if Maple finds parameter values for which the pair S, M is naturally reductive and false otherwise; Eq is the set of equations (with the variables parm as unknowns) which must be satisfied for S, M to be naturally reductive; Soln is the list of solutions to the equations Eq; and NatRedPair is the list of naturally reductive subspaces and inner products obtained from the parameter values given by the different solutions in 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.
>
|
|
>
|
|
Define a subspace S1, a complement M1, and an inner product B1 on M1.
| (2.1) |
Check that S1, M1 are naturally reductive with respect to B1.
Alg1 >
|
|
| (2.2) |
Naturally reductive means that the symmetric tensor g defined by B is invariant with respect to the vectors in S and the Lie derivative of g with respect to the vectors in M vanishes on pairs of vectors from M. Thus, for the above example we have:
Alg1 >
|
|
| (2.3) |
Alg1 >
|
|
| (2.4) |
Alg1 >
|
|
| (2.5) |
Alg1 >
|
|
| (2.6) |
Example 2.
In this example we consider a Lie algebra containing a parameter b. We find that a certain subspace S2 admits a naturally reductive complement M2 when b = 0.
First initialize a Lie algebra and display the Lie bracket multiplication table.
Alg1 >
|
|
| (2.7) |
Alg1 >
|
|
For S2 we have that M2 is a reductive complement. We let the inner product B2 be arbitrary.
| (2.8) |
Alg2 >
|
|
| (2.9) |
Alg2 >
|
|
| (2.10) |
We see that [e2, e3] is naturally reductive only when b = 0. To check this we substitute b = 0 into the Lie algebra data structure for L2 and change the name of the algebra to Alg3.
Alg2 >
|
|
| (2.11) |
Alg2 >
|
|
| (2.12) |
Alg2 >
|
|
| (2.14) |
|
|