 DifferentialGeometry/LieAlgebras/Query/CartanInvolution - Maple Help

Query[CartanInvolution] - check if a linear transformation of a semi-simple, real Lie algebra is a Cartan involution

Calling Sequences

Query(

Parameters

Theta    - a transformation, mapping a semi-simple Lie algebra to itself 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 involution of g is a Lie algebra automorphism Θ : g → g such that [i], and [ii] the symmetric bilinear form is positive-definite. Examples

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

We check to see if some transformations of are Cartan involutions. 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)

Define a transformation and check that it is an involution.

 sl2 > $\mathrm{Θ1}≔\mathrm{Transformation}\left(\left[\left[\mathrm{e1},-\mathrm{e1}\right],\left[\mathrm{e2},-\mathrm{e3}\right],\left[\mathrm{e3},-\mathrm{e2}\right]\right]\right)$
 ${\mathrm{Θ1}}{:=}\left[\left[{\mathrm{e1}}{,}{-}{\mathrm{e1}}\right]{,}\left[{\mathrm{e2}}{,}{-}{\mathrm{e3}}\right]{,}\left[{\mathrm{e3}}{,}{-}{\mathrm{e2}}\right]\right]$ (2.3)
 sl2 > $\mathrm{Query}\left(\mathrm{Θ1},"CartanInvolution"\right)$
 ${\mathrm{true}}$ (2.4)

Define a transformation It is a homomorphism, , but the symmetric bilinear form is not positive-definite.

 sl2 > $\mathrm{Θ2}≔\mathrm{Transformation}\left(\left[\left[\mathrm{e1},\mathrm{e1}\right],\left[\mathrm{e2},-\mathrm{e2}\right],\left[\mathrm{e3},-\mathrm{e3}\right]\right]\right)$
 ${\mathrm{Θ2}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e1}}\right]{,}\left[{\mathrm{e2}}{,}{-}{\mathrm{e2}}\right]{,}\left[{\mathrm{e3}}{,}{-}{\mathrm{e3}}\right]\right]$ (2.5)
 sl2 > $\mathrm{Query}\left(\mathrm{Θ2},"CartanInvolution"\right)$
 ${\mathrm{false}}$ (2.6)

The map is a homomorphism.

 sl2 > $\mathrm{Query}\left(\mathrm{Θ2},"Homomorphism"\right)$
 ${\mathrm{true}}$ (2.7)

The map satisfies ,

 sl2 > $\mathrm{ComposeTransformations}\left(\mathrm{Θ2},\mathrm{Θ2}\right)$
 $\left[\left[{\mathrm{e1}}{,}{\mathrm{e1}}\right]{,}\left[{\mathrm{e2}}{,}{\mathrm{e2}}\right]{,}\left[{\mathrm{e3}}{,}{\mathrm{e3}}\right]\right]$ (2.8)

The symmetric bilinear form is not positive-definite.

 sl2 > $V≔\left[\mathrm{e1},\mathrm{e2},\mathrm{e3}\right]$
 ${V}{:=}\left[{\mathrm{e1}}{,}{\mathrm{e2}}{,}{\mathrm{e3}}\right]$ (2.9)
 sl2 > $\mathrm{Matrix}\left(3,3,\left(i,j\right)↦\mathrm{Killing}\left(V\left[i\right],\mathrm{ApplyHomomorphism}\left(\mathrm{Θ2},V\left[j\right]\right)\right)\right)$
 $\left[\begin{array}{rrr}{8}& {0}& {0}\\ {0}& {0}& {-}{4}\\ {0}& {-}{4}& {0}\end{array}\right]$ (2.10) See Also