ReductivePair - Maple Help

Query[ReductivePair] - check if a subalgebra, subspace pair defines a reductive pair in a Lie algebra

Calling Sequences

Query(S, M, "ReductivePair")

Query(S, M, parm, "ReductivePair")

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 complementary subspace to S in g

parm    - (optional) a set of parameters appearing in the list of vectors S

Description

 • Let be a Lie algebra, a subalgebra and a subspace. Then the subalgebra, subspace pair is called a reductive pair if [i] (vector space direct sum) and [ii]  for all and y in $M$. The subspace is called a reductive complement for the subalgebra $S$.
 • Query(S, M, "ReductivePair") returns true if the subspace M defines a reductive complement to the subalgebra S.
 • Query(S, M, parm, "ReductivePair") returns a sequence TF, Eq, Soln, reductiveList. Here TF is true if Maple finds parameter values for which M is a reductive complement and false otherwise; Eq is the set of equations (with the variables parm as unknowns) which must be satisfied for M to be a reductive complement; Soln is the list of solutions to the equations Eq; and reductiveList is the list of reductive subspaces 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.

 > $L≔\mathrm{_DG}\left(\left[\left["LieAlgebra",\mathrm{Alg},\left[4\right]\right],\left[\left[\left[1,4,1\right],0\right],\left[\left[2,3,1\right],1\right],\left[\left[2,4,2\right],1\right],\left[\left[3,4,3\right],-1\right]\right]\right]\right)$
 ${L}{:=}\left[\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e3}}\right]$ (2.1)
 > $\mathrm{DGsetup}\left(L\right):$

We see that span is not a reductive complement for span but span is a reductive complement for span

 Alg > $\mathrm{S1}≔\left[\mathrm{e1},\mathrm{e2}\right]:$$\mathrm{M1}≔\left[\mathrm{e3},\mathrm{e4}\right]:$
 Alg > $\mathrm{Query}\left(\mathrm{S1},\mathrm{M1},"ReductivePair"\right)$
 ${\mathrm{false}}$ (2.2)
 Alg > $\mathrm{S2}≔\left[\mathrm{e3},\mathrm{e4}\right]:$$\mathrm{M2}≔\left[\mathrm{e1},\mathrm{e2}\right]:$
 Alg > $\mathrm{Query}\left(\mathrm{S2},\mathrm{M2},"ReductivePair"\right)$
 ${\mathrm{true}}$ (2.3)

Now we look for the most general reductive complement ${M}_{3}$ for the subalgebra span .

 Alg > $\mathrm{S3}≔\left[\mathrm{e3},\mathrm{e4}\right]:$$\mathrm{M3}≔\mathrm{evalDG}\left(\left[\mathrm{e1}+\mathrm{a1}\mathrm{e3}+\mathrm{a2}\mathrm{e4},\mathrm{e2}+\mathrm{a3}\mathrm{e3}+\mathrm{a4}\mathrm{e4}\right]\right):$
 Alg > $\mathrm{TF},\mathrm{EQ},\mathrm{SOL},\mathrm{redPair}≔\mathrm{Query}\left(\mathrm{S3},\mathrm{M3},\left\{\mathrm{a1},\mathrm{a2},\mathrm{a3},\mathrm{a4}\right\},"ReductivePair"\right)$
 ${\mathrm{TF}}{,}{\mathrm{EQ}}{,}{\mathrm{SOL}}{,}{\mathrm{redPair}}{:=}{\mathrm{true}}{,}\left\{{0}{,}{\mathrm{a2}}{,}{-}{\mathrm{a1}}{,}{-}{\mathrm{a2}}{,}{-}{2}{}{\mathrm{a3}}{,}{-}{\mathrm{a4}}{,}{-}{\mathrm{a1}}{+}{\mathrm{a4}}\right\}{,}\left[\left\{{\mathrm{a1}}{=}{0}{,}{\mathrm{a2}}{=}{0}{,}{\mathrm{a3}}{=}{0}{,}{\mathrm{a4}}{=}{0}\right\}\right]{,}\left[\left[\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{,}\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]\right]\right]$ (2.4)

The only possibility is span.

 Alg > $\mathrm{redPair}\left[1\right]$
 $\left[\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{,}\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]\right]$ (2.5)

Note that the ComplementaryBasis command can be used to generate the most general complementary subspace. This helps to calculate reductive complements for subalgebras.

 Alg > $\mathrm{S4}≔\left[\mathrm{e4}\right]:$
 Alg > $\mathrm{M4}≔\mathrm{ComplementaryBasis}\left(\mathrm{S4},\left[\mathrm{e1},\mathrm{e2},\mathrm{e3},\mathrm{e4}\right],a\right)$
 ${\mathrm{M4}}{:=}\left[{\mathrm{e1}}{+}{\mathrm{a1}}{}{\mathrm{e4}}{,}{\mathrm{e2}}{+}{\mathrm{a2}}{}{\mathrm{e4}}{,}{\mathrm{e3}}{+}{\mathrm{a3}}{}{\mathrm{e4}}\right]{,}\left\{{\mathrm{a1}}{,}{\mathrm{a2}}{,}{\mathrm{a3}}\right\}$ (2.6)
 Alg > $\mathrm{Query}\left(\mathrm{S4},\mathrm{M4},"ReductivePair"\right)$
 ${\mathrm{true}}{,}\left\{{0}{,}{\mathrm{a3}}{,}{-}{\mathrm{a2}}\right\}{,}\left[\left\{{\mathrm{a1}}{=}{\mathrm{a1}}{,}{\mathrm{a2}}{=}{0}{,}{\mathrm{a3}}{=}{0}\right\}\right]{,}\left[\left[\left[{\mathrm{e4}}\right]{,}\left[{\mathrm{e1}}{+}{\mathrm{a1}}{}{\mathrm{e4}}{,}{\mathrm{e2}}{,}{\mathrm{e3}}\right]\right]\right]$ (2.7)