find the compact roots in a root system for a non-compact semi-simple real Lie algebra - Maple Programming Help

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LieAlgebras[CompactRoots] - find the compact roots in a root system for a non-compact semi-simple real Lie algebra

Calling Sequences

CompactRoots(A, CSA)

Parameters

$\mathrm{Δ}$       - a list of column vectors, defining the root system, positive roots or simple roots of a non-compact semi-simple Lie algebra

A       - a list of vectors in a Lie algebra, defining a subalgebra of the Cartan subalgebra on which the Killing form is negative-definite

  CSA     - a list of vectors, defining the Cartan subalgebra of a non-compact semi-simple Lie algebra



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.
 • Every non-compact semi-simple real Lie algebra g admits a Cartan decomposition g = t ⊕p . Here t is a subalgebra, p a subspace, [t, p] ⊆ p and [p, p] ⊆ t, that is, t and p define a symmetric pair. Moreover, the Killing form is negative-definite on t and positive-definite on p.
 • Let h be a Cartan subalgebra for g and let be the associated root system. Set a = h ⋂ p. Then the set of compact roots is defined to be

This means that if we choose a basis for a and extend to a basis  for h, then the components of a compact root ${\mathrm{α}}_{}$ in the directions are 0. If  determines the root space for then  for  With respect to the standard Cartan algebras for the non-compact, simple matrix algebras we consider here, the compact roots are precisely those which are purely imaginary complex numbers.

 • In the Satake diagram for a non-compact semi-simple real Lie algebra, the compact roots are given a different color from the other roots.

Examples

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

Example 1.

We find the compact roots for First we use the command SimpleLieAlgebraData to initialize the Lie algebra

 > $\mathrm{LD}≔\mathrm{SimpleLieAlgebraData}\left("su\left(5, 2\right)",\mathrm{su52},\mathrm{labelformat}="gl",\mathrm{labels}=\left['E','\mathrm{θ}'\right]\right):$
 > $\mathrm{DGsetup}\left(\mathrm{LD}\right)$
 ${\mathrm{Lie algebra: su52}}$ (2.1)

For this example we use the command SimpleLieAlgebraProperties to generate the various properties of that we need.

 su52 > $\mathrm{Properties_su52}≔\mathrm{SimpleLieAlgebraProperties}\left(\mathrm{su52}\right):$

Here is the Cartan subalgebra.

 su52 > $\mathrm{CSA}≔{\mathrm{Properties_su52}}_{"CartanSubalgebra"}$
 ${\mathrm{CSA}}{:=}\left[{\mathrm{E11}}{,}{\mathrm{E22}}{,}{\mathrm{Ei11}}{,}{\mathrm{Ei22}}{,}{\mathrm{Ei55}}{,}{\mathrm{Ei66}}\right]$ (2.2)

Here is the Cartan subalgebra decomposition

 su52 > $T,A≔{\mathrm{Properties_su52}}_{"CartanSubalgebraDecomposition"}$
 ${T}{,}{A}{:=}\left[{\mathrm{Ei11}}{,}{\mathrm{Ei22}}{,}{\mathrm{Ei55}}{,}{\mathrm{Ei66}}\right]{,}\left[{\mathrm{E11}}{,}{\mathrm{E22}}\right]$ (2.3)

We check that the restriction of the Killing form to the diagonal matrices  with imaginary entries is negative-definite. The restriction of the Killing form to the diagonal matrices $A$ with real entries is positive-definite.

 su52 > $\mathrm{K1}≔\mathrm{Killing}\left(T\right)$
 ${\mathrm{K1}}{:=}\left[\begin{array}{rrrr}{-}{56}& {-}{28}& {0}& {14}\\ {-}{28}& {-}{56}& {0}& {14}\\ {0}& {0}& {-}{28}& {-}{14}\\ {14}& {14}& {-}{14}& {-}{28}\end{array}\right]$ (2.4)
 su52 > $\mathrm{LinearAlgebra}:-\mathrm{IsDefinite}\left(\mathrm{K1},\mathrm{query}='\mathrm{negative_definite}'\right)$
 ${\mathrm{true}}$ (2.5)
 su52 > $\mathrm{K2}≔\mathrm{Killing}\left(A\right)$
 ${\mathrm{K2}}{:=}\left[\begin{array}{rr}{28}& {0}\\ {0}& {28}\end{array}\right]$ (2.6)

The second list of vectors in (2.3)  is therefore our subalgebra as described above.  Next we find the positive roots.

 su52 > $\mathrm{PT}≔{\mathrm{Properties_su52}}_{"PositiveRoots"}$
 ${\mathrm{PT}}{:=}\left[\left[\begin{array}{c}{1}\\ {-}{1}\\ {-}{I}\\ {I}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {1}\\ {-}{I}\\ {-}{2}{}{I}\\ {-}{I}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {0}\\ {I}\\ {I}\\ {I}\\ {-}{I}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {0}\\ {-}{I}\\ {-}{I}\\ {I}\\ {2}{}{I}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {1}\\ {I}\\ {2}{}{I}\\ {-}{I}\\ {-}{I}\end{array}\right]{,}\left[\begin{array}{c}{1}\\ {-}{1}\\ {I}\\ {-}{I}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{1}\\ {0}\\ {-}{2}{}{I}\\ {-}{I}\\ {-}{I}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {1}\\ {0}\\ {-}{I}\\ {0}\\ {-}{I}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {0}\\ {0}\\ {0}\\ {2}{}{I}\\ {I}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {1}\\ {0}\\ {I}\\ {0}\\ {I}\end{array}\right]{,}\left[\begin{array}{c}{1}\\ {0}\\ {2}{}{I}\\ {I}\\ {-}{I}\\ {-}{I}\end{array}\right]{,}\left[\begin{array}{c}{1}\\ {0}\\ {-}{I}\\ {0}\\ {0}\\ {-}{I}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {1}\\ {-}{I}\\ {-}{2}{}{I}\\ {I}\\ {I}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {1}\\ {I}\\ {2}{}{I}\\ {I}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{1}\\ {0}\\ {I}\\ {0}\\ {0}\\ {I}\end{array}\right]{,}\left[\begin{array}{c}{1}\\ {0}\\ {-}{2}{}{I}\\ {-}{I}\\ {I}\\ {I}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {2}\\ {0}\\ {0}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{1}\\ {0}\\ {2}{}{I}\\ {I}\\ {I}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{1}\\ {1}\\ {-}{I}\\ {I}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{1}\\ {1}\\ {I}\\ {-}{I}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{2}\\ {0}\\ {0}\\ {0}\\ {0}\\ {0}\end{array}\right]\right]$ (2.7)

The compact roots are:

 su52 > $\mathrm{CompactRoots}\left(\mathrm{PT},A,\mathrm{CSA}\right)$
 $\left[\left[\begin{array}{c}{0}\\ {0}\\ {I}\\ {I}\\ {I}\\ {-}{I}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {0}\\ {-}{I}\\ {-}{I}\\ {I}\\ {2}{}{I}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {0}\\ {0}\\ {0}\\ {2}{}{I}\\ {I}\end{array}\right]\right]$ (2.8)
 $\left[\left[\begin{array}{c}{2}{}{I}\\ {2}{}{I}\\ {-}{2}{}{I}\\ {I}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {0}\\ {I}\\ {-}{2}{}{I}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{2}{}{I}\\ {2}{}{I}\\ {-}{I}\\ {-}{I}\\ {0}\\ {0}\end{array}\right]\right]$ (2.9)

Note that these roots all have purely imaginary components.

Example 2.

We find the compact roots for First we use the command SimpleLieAlgebraData to initialize the Lie algebra

 > $\mathrm{LD}≔\mathrm{SimpleLieAlgebraData}\left("sp\left(4, 4\right)",\mathrm{sp44},\mathrm{labelformat}="gl",\mathrm{labels}=\left['S','\mathrm{σ}'\right]\right):$
 > $\mathrm{DGsetup}\left(\mathrm{LD}\right)$
 ${\mathrm{Lie algebra: sp44}}$ (2.10)

We use the command SimpleLieAlgebraProperties to generate the various properties of that we need.

 sp44 > $\mathrm{Properties_sp44}≔\mathrm{SimpleLieAlgebraProperties}\left(\mathrm{sp44}\right):$

Here is the Cartan subalgebra.

 sp44 > $\mathrm{CSA}≔{\mathrm{Properties_sp44}}_{"CartanSubalgebra"}$
 ${\mathrm{CSA}}{:=}\left[{\mathrm{S13}}{,}{\mathrm{S24}}{,}{\mathrm{Si11}}{+}{\mathrm{Si33}}{,}{\mathrm{Si22}}{+}{\mathrm{Si44}}\right]$ (2.11)

Here is the Cartan subalgebra decomposition

 sp44 > $T,A≔{\mathrm{Properties_sp44}}_{"CartanSubalgebraDecomposition"}$
 ${T}{,}{A}{:=}\left[{\mathrm{Si11}}{+}{\mathrm{Si33}}{,}{\mathrm{Si22}}{+}{\mathrm{Si44}}\right]{,}\left[{\mathrm{S13}}{,}{\mathrm{S24}}\right]$ (2.12)

The restriction of the Killing form to the diagonal matrices with imaginary entries is negative-definite. The restriction of the Killing form to the diagonal matrices $A$ with real entries is positive-definite.

 sp44 > $\mathrm{K1}≔\mathrm{Killing}\left(T\right)$
 ${\mathrm{K1}}{:=}\left[\begin{array}{rr}{-}{40}& {0}\\ {0}& {-}{40}\end{array}\right]$ (2.13)
 sp44 > $\mathrm{LinearAlgebra}:-\mathrm{IsDefinite}\left(\mathrm{K1},\mathrm{query}='\mathrm{negative_definite}'\right)$
 ${\mathrm{true}}$ (2.14)
 sp44 > $\mathrm{K2}≔\mathrm{Killing}\left(A\right)$
 ${\mathrm{K2}}{:=}\left[\begin{array}{rr}{40}& {0}\\ {0}& {40}\end{array}\right]$ (2.15)

The second list of vectors in (2.3)  is therefore our subalgebra as described above.

Next we find the positive roots.

 sp44 > $\mathrm{PT}≔{\mathrm{Properties_sp44}}_{"PositiveRoots"}$
 ${\mathrm{PT}}{:=}\left[\left[\begin{array}{c}{0}\\ {0}\\ {2}{}{I}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{1}\\ {-}{1}\\ {-}{I}\\ {-}{I}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {0}\\ {0}\\ {2}{}{I}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {2}\\ {0}\\ {-}{2}{}{I}\end{array}\right]{,}\left[\begin{array}{c}{1}\\ {-}{1}\\ {I}\\ {-}{I}\end{array}\right]{,}\left[\begin{array}{c}{1}\\ {-}{1}\\ {-}{I}\\ {I}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {2}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{1}\\ {-}{1}\\ {I}\\ {I}\end{array}\right]{,}\left[\begin{array}{c}{1}\\ {1}\\ {-}{I}\\ {-}{I}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {2}\\ {0}\\ {2}{}{I}\end{array}\right]{,}\left[\begin{array}{c}{1}\\ {1}\\ {I}\\ {-}{I}\end{array}\right]{,}\left[\begin{array}{c}{1}\\ {1}\\ {-}{I}\\ {I}\end{array}\right]{,}\left[\begin{array}{c}{1}\\ {1}\\ {I}\\ {I}\end{array}\right]{,}\left[\begin{array}{c}{2}\\ {0}\\ {-}{2}{}{I}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{2}\\ {0}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{2}\\ {0}\\ {2}{}{I}\\ {0}\end{array}\right]\right]$ (2.16)

The compact roots are:

 sp44 > $\mathrm{CompactRoots}\left(\mathrm{PT},A,\mathrm{CSA}\right)$
 $\left[\left[\begin{array}{c}{0}\\ {0}\\ {2}{}{I}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {0}\\ {0}\\ {2}{}{I}\end{array}\right]\right]$ (2.17)