 SymmetricPair - Maple Help

Query[SymmetricPair] - check if a subalgebra, subspace pair defines a symmetric pair in a Lie algebra

Calling Sequences

Query(M, S, "SymmetricPair")

Query(M, S, parm, "SymmetricPair")

Parameters

M      - a list of independent vectors which defines a reductive complement to S in a Lie algebra $\mathrm{𝔤}$

S      - a list of independent vectors which form a subalgebra in $\mathrm{𝔤}$

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

 • Let be a Lie algebra,a subalgebra, and  a subspace. The subalgebra, subspace pair  is a symmetric pair if [i] [ii] for and and [iii]  for and . Note that [i] and [ii] imply that define a  reductive pair. If  is a symmetric pair then  is a naturally reductive pair for any inner product on If  is a symmetric pair, then  is called a symmetric complement to the subalgebra $S$.
 • Query(M, S, "SymmetricPair") returns true if the subspace M defines a symmetric complement to the subalgebra S, and false otherwise.
 • Query(M, S, parm, "SymmetricPair") returns a sequence TF, Eq, Soln, symmetricList. Here TF is true if Maple finds parameter values for which S is a symmetric complement and false otherwise; Eq is the set of equations (with the variables parm as unknowns) which must be satisfied for S to be a symmetric complement; Soln is the list of solutions to the equations Eq; and symmetricList is the list of symmetric complements 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 and display the Lie bracket multiplication table.

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

We can check that the subspace span defines a symmetric complement for the subalgebra span.

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

In fact, we can show that span is the only symmetric complement to by constructing the general complement span.

 Alg > $\mathrm{S2}≔\left[\mathrm{e1},\mathrm{e4}\right]:$$\mathrm{M2}≔\left[\mathrm{e2}+\mathrm{a1}\mathrm{e1}+\mathrm{a2}\mathrm{e4},\mathrm{e3}+\mathrm{a3}\mathrm{e1}+\mathrm{a4}\mathrm{e4}\right]:$
 Alg > $\mathrm{TF},\mathrm{EQ},\mathrm{SOLN},\mathrm{symPair}≔\mathrm{Query}\left(\mathrm{S2},\mathrm{M2},\left\{\mathrm{a1},\mathrm{a2},\mathrm{a3},\mathrm{a4}\right\},"ReductivePair"\right)$
 ${\mathrm{TF}}{,}{\mathrm{EQ}}{,}{\mathrm{SOLN}}{,}{\mathrm{symPair}}{:=}{\mathrm{true}}{,}\left\{{0}{,}{\mathrm{a3}}{,}{\mathrm{a4}}{,}{-}{\mathrm{a1}}{,}{-}{\mathrm{a2}}\right\}{,}\left[\left\{{\mathrm{a1}}{=}{0}{,}{\mathrm{a2}}{=}{0}{,}{\mathrm{a3}}{=}{0}{,}{\mathrm{a4}}{=}{0}\right\}\right]{,}\left[\left[\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]\right]\right]$ (2.3)

SOLN shows that all the parameters  must be zero in order for to define a symmetric pair.

 Alg > $\mathrm{SOLN}$
 $\left[\left\{{\mathrm{a1}}{=}{0}{,}{\mathrm{a2}}{=}{0}{,}{\mathrm{a3}}{=}{0}{,}{\mathrm{a4}}{=}{0}\right\}\right]$ (2.4)

Next we show that the subalgebra  spandoes not admit a symmetric complement at all.

 Alg > $\mathrm{S3}≔\left[\mathrm{e4}\right]:$
 Alg > $\mathrm{M3}≔\mathrm{evalDG}\left(\left[\mathrm{e1}+\mathrm{a1}\mathrm{e4},\mathrm{e2},\mathrm{e3}\right]\right):$
 Alg > $\mathrm{Query}\left(\mathrm{S3},\mathrm{M3},"ReductivePair"\right)$
 ${\mathrm{true}}$ (2.5)
 Alg > $\mathrm{Query}\left(\mathrm{S3},\mathrm{M3},\left\{\mathrm{a1}\right\},"SymmetricPair"\right)$
 ${\mathrm{false}}{,}\left\{{0}{,}{1}{,}{\mathrm{a1}}{,}{-}{\mathrm{a1}}\right\}$ (2.6)