check if a subalgebra, subspace pair is naturally reductive with respect to an inner product on the subspace - Maple Programming Help

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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

 • Let be a Lie algebra,  a subalgebra, and  a subspace. Let $B$ be a non-degenerate inner product on $M$.  Then the subalgebra, subspace pair  is called naturally reductive with respect to the inner product if [i] the subspace $M$ defines a reductive complement to the subalgebra $S,$ and [ii] the inner product $B$ is  invariant, that is, for all  and .  Here ${\left[x,y\right]}_{M}$ denotes the $M-$component of with respect to the decomposition .
 • 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

 > $\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):$$\mathrm{with}\left(\mathrm{LieAlgebras}\right):$

Example 1.

First initialize a Lie algebra.

 > $\mathrm{L1}≔\mathrm{_DG}\left(\left[\left["LieAlgebra",\mathrm{Alg1},\left[3\right]\right],\left[\left[\left[1,2,1\right],1\right],\left[\left[1,3,2\right],-2\right],\left[\left[2,3,3\right],1\right]\right]\right]\right):$
 > $\mathrm{DGsetup}\left(\mathrm{L1}\right):$

Define a subspace ${S}_{1}$, a complement ${M}_{1}$, and an inner product ${B}_{1}$ on ${M}_{1}$.

 Alg1 > $\mathrm{S1}≔\left[\mathrm{e2}\right]:$$\mathrm{M1}≔\left[\mathrm{e1},\mathrm{e3}\right]:$$\mathrm{B1}≔\mathrm{Matrix}\left(\left[\left[0,1\right],\left[1,0\right]\right]\right)$
 ${\mathrm{B1}}{:=}\left[\begin{array}{rr}{0}& {1}\\ {1}& {0}\end{array}\right]$ (2.1)

Check that ${S}_{1}$, is naturally reductive with respect to ${B}_{1}.$

 Alg1 > $\mathrm{Query}\left(\mathrm{S1},\mathrm{M1},\mathrm{B1},"NaturallyReductivePair"\right)$
 ${\mathrm{true}}$ (2.2)

Naturally reductive means that [i] the symmetric tensor $g$ defined by is invariant with respect to the vectors in ${S}_{1}$ and [ii] the Lie derivative of $B$with respect to the vectors in vanishes on pairs of vectors from ${M}_{1}$. Thus, for the above example we have:

 Alg1 > $g≔\mathrm{evalDG}\left(\mathrm{θ1}&t\mathrm{θ3}+\mathrm{θ3}&t\mathrm{θ1}\right)$
 ${g}{:=}{\mathrm{θ1}}{}{\mathrm{θ3}}{+}{\mathrm{θ3}}{}{\mathrm{θ1}}$ (2.3)
 Alg1 > $\mathrm{LieDerivative}\left(\mathrm{e2},g\right)$
 ${0}{}{\mathrm{θ1}}{}{\mathrm{θ1}}$ (2.4)
 Alg1 > $\mathrm{LieDerivative}\left(\mathrm{e1},g\right)$
 ${-}{\mathrm{θ2}}{}{\mathrm{θ3}}{-}{\mathrm{θ3}}{}{\mathrm{θ2}}$ (2.5)
 Alg1 > $\mathrm{LieDerivative}\left(\mathrm{e3},g\right)$
 ${\mathrm{θ1}}{}{\mathrm{θ2}}{+}{\mathrm{θ2}}{}{\mathrm{θ1}}$ (2.6)

Example 2.

In this example we consider a Lie algebra containing a parameter $b$.  We find that a certain subspace admits a naturally reductive complement ${M}_{2}$ when

First initialize a Lie algebra and display the Lie bracket multiplication table.

 Alg1 > $\mathrm{L2}≔\mathrm{_DG}\left(\left[\left["LieAlgebra",\mathrm{Alg2},\left[4\right]\right],\left[\left[\left[1,4,1\right],1\right],\left[\left[2,4,2\right],b\right],\left[\left[3,4,2\right],1\right],\left[\left[2,4,3\right],-1\right],\left[\left[3,4,3\right],b\right]\right]\right]\right)$
 ${\mathrm{L2}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{b}{}{\mathrm{e2}}{-}{\mathrm{e3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}{+}{b}{}{\mathrm{e3}}\right]$ (2.7)
 Alg1 > $\mathrm{DGsetup}\left(\mathrm{L2}\right):$

For we have that ${M}_{2}$ is a reductive complement.  We let the inner product ${B}_{2}$ be arbitrary.

 Alg2 > $\mathrm{S2}≔\left[\mathrm{e1},\mathrm{e4}\right]:$$\mathrm{M2}≔\left[\mathrm{e2},\mathrm{e3}\right]:$$\mathrm{B2}≔\mathrm{Matrix}\left(\left[\left[r,s\right],\left[s,t\right]\right]\right)$
 ${\mathrm{B2}}{:=}\left[\begin{array}{cc}{r}& {s}\\ {s}& {t}\end{array}\right]$ (2.8)
 Alg2 > $\mathrm{Query}\left(\mathrm{S2},\mathrm{M2},\mathrm{B2},"NaturallyReductivePair"\right)$
 ${\mathrm{false}}$ (2.9)
 Alg2 > $\mathrm{TF},\mathrm{EQ},\mathrm{SOLN},\mathrm{natRedPair}≔\mathrm{Query}\left(\mathrm{S2},\mathrm{M2},\mathrm{B2},\left\{b,r,s,t\right\},"NaturallyReductivePair"\right)$
 ${\mathrm{TF}}{,}{\mathrm{EQ}}{,}{\mathrm{SOLN}}{,}{\mathrm{natRedPair}}{:=}{\mathrm{true}}{,}\left\{{0}{,}{-}{2}{}{s}{-}{2}{}{b}{}{t}{,}{-}{2}{}{b}{}{r}{+}{2}{}{s}{,}{-}{2}{}{b}{}{s}{-}{r}{+}{t}\right\}{,}\left[\left\{{b}{=}{0}{,}{r}{=}{t}{,}{s}{=}{0}{,}{t}{=}{t}\right\}{,}\left\{{b}{=}{b}{,}{r}{=}{0}{,}{s}{=}{0}{,}{t}{=}{0}\right\}{,}\left\{{b}{=}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{1}\right){,}{r}{=}{-}{t}{,}{s}{=}{-}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{1}\right){}{t}{,}{t}{=}{t}\right\}\right]{,}\left[\left[\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{,}\left[\begin{array}{cc}{t}& {0}\\ {0}& {t}\end{array}\right]\right]\right]$ (2.10)

We see that the that span  is naturally reductive only when .  To check this we substitute into the Lie algebra data structure for L2 and change the name of the algebra to Alg3.

 Alg2 > $\mathrm{L3}≔\mathrm{subs}\left(b=0,\mathrm{L2}\right)$
 ${\mathrm{L3}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}\right]$ (2.11)
 Alg2 > $\mathrm{DGsetup}\left(\mathrm{L3}\right)$
 ${\mathrm{Lie algebra: Alg2}}$ (2.12)
 Alg2 > $\mathrm{S3}≔{{\mathrm{natRedPair}}_{1}}_{1};$$\mathrm{M3}≔{{\mathrm{natRedPair}}_{1}}_{2};$$\mathrm{B3}≔{{\mathrm{natRedPair}}_{1}}_{3}$
 ${\mathrm{S3}}{:=}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]$
 ${\mathrm{M3}}{:=}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]$
 ${\mathrm{B3}}{:=}\left[\begin{array}{cc}{t}& {0}\\ {0}& {t}\end{array}\right]$ (2.13)
 Alg2 > $\mathrm{Query}\left(\mathrm{S3},\mathrm{M3},\mathrm{B3},"NaturallyReductivePair"\right)$
 ${\mathrm{true}}$ (2.14)