 DifferentialGeometry/LieAlgebras/Query/CartanDecomposition - Maple Help

Query[CartanDecomposition] - check that two subspaces in a Lie algebra define a Cartan decomposition.

Calling Sequences

Query()

Parameters

T   - a list of vectors, defining a subalgebra of a Lie algebra on which the Killing form is negative-definite.

P   - a list of vectors, defining a subspace of a Lie algebra on which the Killing form is positive-definite Description

 • Let g be a semi-simple real Lie algebra. Then g is called compact if the Killing form  of g is negative-definite, otherwise  g is called non-compact.
 • A Cartan decomposition is a vector space decomposition g = t ⊕p , where [i] t is a subalgebra, [ii] p is a subspace, [iii] [t, p] ⊆ p, [iv] [p, p] ⊆ t, [v] the Killing form is negative-definite on t and [vi] Killing form is positive-definite on p. Examples

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

Example 1.

We check to see if some decompositions of are Cartan decompositions. Initialize the Lie algebra $\mathrm{sl}\left(2\right)$.

 > $\mathrm{LD}≔\mathrm{LieAlgebraData}\left(\left[\left[h,x\right]=2x,\left[h,y\right]=-2y,\left[x,y\right]=h\right],\left[h,x,y\right],\mathrm{sl2}\right)$
 ${\mathrm{LD}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{2}{}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{-}{2}{}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}\right]$ (2.1)
 > $\mathrm{DGsetup}\left(\mathrm{LD}\right)$
 ${\mathrm{Lie algebra: sl2}}$ (2.2)

The decomposition gives a Cartan decomposition.

 > $\mathrm{T1},\mathrm{P1}≔\mathrm{evalDG}\left(\left[\mathrm{e2}-\mathrm{e3}\right]\right),\mathrm{evalDG}\left(\left[\mathrm{e1},\mathrm{e2}+\mathrm{e3}\right]\right)$
 ${\mathrm{T1}}{,}{\mathrm{P1}}{:=}\left[{\mathrm{e2}}{-}{\mathrm{e3}}\right]{,}\left[{\mathrm{e1}}{,}{\mathrm{e2}}{+}{\mathrm{e3}}\right]$ (2.3)
 > $\mathrm{Query}\left(\mathrm{T1},\mathrm{P1},"CartanDecomposition"\right)$
 ${\mathrm{true}}$ (2.4)

The decomposition gives a symmetric pair but not a Cartan decomposition.

 > $\mathrm{T2},\mathrm{P2}≔\mathrm{evalDG}\left(\left[\mathrm{e2}+\mathrm{e3}\right]\right),\mathrm{evalDG}\left(\left[\mathrm{e1},\mathrm{e2}-\mathrm{e3}\right]\right)$
 ${\mathrm{T2}}{,}{\mathrm{P2}}{:=}\left[{\mathrm{e2}}{+}{\mathrm{e3}}\right]{,}\left[{\mathrm{e1}}{,}{\mathrm{e2}}{-}{\mathrm{e3}}\right]$ (2.5)
 > $\mathrm{Query}\left(\mathrm{T2},\mathrm{P2},"CartanDecomposition"\right)$
 ${\mathrm{false}}$ (2.6)
 sl2 > $\mathrm{Query}\left(\mathrm{T2},\mathrm{P2},"SymmetricPair"\right)$
 ${\mathrm{true}}$ (2.7)
 sl2 > $\mathrm{Killing}\left(\mathrm{P2}\right)$
 $\left[\begin{array}{rr}{8}& {0}\\ {0}& {-}{8}\end{array}\right]$ (2.8)