associate to each positive root of a simple Lie algebra a vector in the Cartan subalgebra - Maple Programming Help

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LieAlgebras[RootToCartanSubalgebraElementH] - associate to each positive root of a simple Lie algebra a vector in the Cartan subalgebra

Calling Sequences

RootToCartanSubalgebraElementH(RSD)

Parameters

$\mathrm{α}$     - a vector, defining a positive (or negative) root of a simple Lie algebra

RSD   - a table, defining the root space decomposition of a simple Lie algebra

Description

 • Let g be a simple Lie algebra, h a Cartan subalgebra, andthe root space decomposition of g with respect to h. For each root , there are vectors and  such that

and

These conditions uniquely determine Note that the vectors define the 3-dimensional Lie algebra $\mathrm{sl}\left(2\right)$. The assignment is used to calculate the Cartan matrix for the Lie algebra $\mathrm{𝔤}$.

 • The procedure RootToCartanSubalgebraElementH(RSD) returns the vector ${H}_{{\mathrm{α}}_{}}.$

Examples

 > with(DifferentialGeometry): with(LieAlgebras):

Example 1.

We consider the Lie algebra This is the 24-dimensional real Lie algebra of 6×6 complex matrices $A$ which are trace-free and skew-Hermitian with respect to the quadratic form $Q=\left[\begin{array}{rr}0& {I}_{3}\\ {I}_{3}& 0\end{array}\right]$ . We use the command SimpleLieAlgebraData to initialize this Lie algebra.

 > LD1 := SimpleLieAlgebraData("su(3,3)", su33, labelformat = "gl", labels = ['E', 'omega']):
 > DGsetup(LD1);
 ${\mathrm{Lie algebra: su33}}$ (2.1)

We use the command SimpleLieAlgebraProperties to obtain the Cartan subalgebra, the root space decomposition, and the simple roots.

 su33 > P := SimpleLieAlgebraProperties(su33):

The result is a table. Here is the Cartan subalgebra for

 su33 > CSA := P["CartanSubalgebra"];
 ${\mathrm{CSA}}{:=}\left[{\mathrm{E11}}{,}{\mathrm{E22}}{,}{\mathrm{E33}}{,}{\mathrm{Ei11}}{,}{\mathrm{Ei22}}\right]$ (2.2)

Here is the root space decomposition for $\mathrm{su}\left(3,3\right).$

 su33 > RSD := eval(P["RootSpaceDecomposition"]);
 ${\mathrm{RSD}}{:=}{\mathrm{table}}\left(\left[\left[{1}{,}{0}{,}{1}{,}{-}{2}{}{I}{,}{-}{I}\right]{=}{\mathrm{E16}}{+}{I}{}{\mathrm{Ei16}}{,}\left[{1}{,}{1}{,}{0}{,}{I}{,}{-}{I}\right]{=}{\mathrm{E15}}{-}{I}{}{\mathrm{Ei15}}{,}\left[{-}{1}{,}{0}{,}{1}{,}{-}{2}{}{I}{,}{-}{I}\right]{=}{\mathrm{E31}}{+}{I}{}{\mathrm{Ei31}}{,}\left[{1}{,}{-}{1}{,}{0}{,}{-}{I}{,}{I}\right]{=}{\mathrm{E12}}{-}{I}{}{\mathrm{Ei12}}{,}\left[{0}{,}{0}{,}{-}{2}{,}{0}{,}{0}\right]{=}{\mathrm{Ei63}}{,}\left[{0}{,}{-}{1}{,}{-}{1}{,}{I}{,}{2}{}{I}\right]{=}{\mathrm{E53}}{-}{I}{}{\mathrm{Ei53}}{,}\left[{0}{,}{-}{2}{,}{0}{,}{0}{,}{0}\right]{=}{\mathrm{Ei52}}{,}\left[{-}{1}{,}{-}{1}{,}{0}{,}{I}{,}{-}{I}\right]{=}{\mathrm{E42}}{-}{I}{}{\mathrm{Ei42}}{,}\left[{1}{,}{0}{,}{-}{1}{,}{2}{}{I}{,}{I}\right]{=}{\mathrm{E13}}{+}{I}{}{\mathrm{Ei13}}{,}\left[{0}{,}{1}{,}{-}{1}{,}{-}{I}{,}{-}{2}{}{I}\right]{=}{\mathrm{E23}}{-}{I}{}{\mathrm{Ei23}}{,}\left[{0}{,}{-}{1}{,}{-}{1}{,}{-}{I}{,}{-}{2}{}{I}\right]{=}{\mathrm{E53}}{+}{I}{}{\mathrm{Ei53}}{,}\left[{2}{,}{0}{,}{0}{,}{0}{,}{0}\right]{=}{\mathrm{Ei14}}{,}\left[{0}{,}{-}{1}{,}{1}{,}{I}{,}{2}{}{I}\right]{=}{\mathrm{E32}}{-}{I}{}{\mathrm{Ei32}}{,}\left[{-}{1}{,}{0}{,}{1}{,}{2}{}{I}{,}{I}\right]{=}{\mathrm{E31}}{-}{I}{}{\mathrm{Ei31}}{,}\left[{-}{1}{,}{-}{1}{,}{0}{,}{-}{I}{,}{I}\right]{=}{\mathrm{E42}}{+}{I}{}{\mathrm{Ei42}}{,}\left[{0}{,}{1}{,}{-}{1}{,}{I}{,}{2}{}{I}\right]{=}{\mathrm{E23}}{+}{I}{}{\mathrm{Ei23}}{,}\left[{0}{,}{1}{,}{1}{,}{-}{I}{,}{-}{2}{}{I}\right]{=}{\mathrm{E26}}{+}{I}{}{\mathrm{Ei26}}{,}\left[{1}{,}{0}{,}{-}{1}{,}{-}{2}{}{I}{,}{-}{I}\right]{=}{\mathrm{E13}}{-}{I}{}{\mathrm{Ei13}}{,}\left[{1}{,}{-}{1}{,}{0}{,}{I}{,}{-}{I}\right]{=}{\mathrm{E12}}{+}{I}{}{\mathrm{Ei12}}{,}\left[{-}{1}{,}{0}{,}{-}{1}{,}{2}{}{I}{,}{I}\right]{=}{\mathrm{E43}}{-}{I}{}{\mathrm{Ei43}}{,}\left[{-}{1}{,}{0}{,}{-}{1}{,}{-}{2}{}{I}{,}{-}{I}\right]{=}{\mathrm{E43}}{+}{I}{}{\mathrm{Ei43}}{,}\left[{0}{,}{2}{,}{0}{,}{0}{,}{0}\right]{=}{\mathrm{Ei25}}{,}\left[{1}{,}{0}{,}{1}{,}{2}{}{I}{,}{I}\right]{=}{\mathrm{E16}}{-}{I}{}{\mathrm{Ei16}}{,}\left[{-}{2}{,}{0}{,}{0}{,}{0}{,}{0}\right]{=}{\mathrm{Ei41}}{,}\left[{-}{1}{,}{1}{,}{0}{,}{I}{,}{-}{I}\right]{=}{\mathrm{E21}}{-}{I}{}{\mathrm{Ei21}}{,}\left[{0}{,}{-}{1}{,}{1}{,}{-}{I}{,}{-}{2}{}{I}\right]{=}{\mathrm{E32}}{+}{I}{}{\mathrm{Ei32}}{,}\left[{1}{,}{1}{,}{0}{,}{-}{I}{,}{I}\right]{=}{\mathrm{E15}}{+}{I}{}{\mathrm{Ei15}}{,}\left[{0}{,}{0}{,}{2}{,}{0}{,}{0}\right]{=}{\mathrm{Ei36}}{,}\left[{-}{1}{,}{1}{,}{0}{,}{-}{I}{,}{I}\right]{=}{\mathrm{E21}}{+}{I}{}{\mathrm{Ei21}}{,}\left[{0}{,}{1}{,}{1}{,}{I}{,}{2}{}{I}\right]{=}{\mathrm{E26}}{-}{I}{}{\mathrm{Ei26}}\right]\right)$ (2.3)

Here are the positive roots.

 su33 > PR := P["PositiveRoots"];
 ${\mathrm{PR}}{:=}\left[\left[\begin{array}{c}{1}\\ {-}{1}\\ {0}\\ {I}\\ {-}{I}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {1}\\ {-}{1}\\ {I}\\ {2}{}{I}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {2}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {1}\\ {-}{1}\\ {-}{I}\\ {-}{2}{}{I}\end{array}\right]{,}\left[\begin{array}{c}{1}\\ {-}{1}\\ {0}\\ {-}{I}\\ {I}\end{array}\right]{,}\left[\begin{array}{c}{1}\\ {0}\\ {-}{1}\\ {2}{}{I}\\ {I}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {1}\\ {1}\\ {I}\\ {2}{}{I}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {1}\\ {1}\\ {-}{I}\\ {-}{2}{}{I}\end{array}\right]{,}\left[\begin{array}{c}{1}\\ {0}\\ {-}{1}\\ {-}{2}{}{I}\\ {-}{I}\end{array}\right]{,}\left[\begin{array}{c}{1}\\ {0}\\ {1}\\ {2}{}{I}\\ {I}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {2}\\ {0}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{1}\\ {0}\\ {1}\\ {-}{2}{}{I}\\ {-}{I}\end{array}\right]{,}\left[\begin{array}{c}{1}\\ {1}\\ {0}\\ {I}\\ {-}{I}\end{array}\right]{,}\left[\begin{array}{c}{1}\\ {1}\\ {0}\\ {-}{I}\\ {I}\end{array}\right]{,}\left[\begin{array}{r}{2}\\ {0}\\ {0}\\ {0}\\ {0}\end{array}\right]\right]$ (2.4)

Let us find ${H}_{{\mathrm{α}}_{}},$where ${\mathrm{α}}_{}$ is the first root (2.4)

 su33 > alpha := PR[1];
 ${\mathrm{α}}{:=}\left[\begin{array}{c}{1}\\ {-}{1}\\ {0}\\ {I}\\ {-}{I}\end{array}\right]$ (2.5)
 su33 > H := RootToCartanSubalgebraElementH(alpha,RSD);
 ${H}{:=}{-}\frac{{I}}{{2}}{}{\mathrm{Ei11}}{+}\frac{{I}}{{2}}{}{\mathrm{Ei22}}{+}\frac{{1}}{{2}}{}{\mathrm{E11}}{-}\frac{{1}}{{2}}{}{\mathrm{E22}}$ (2.6)

We check that is in the Cartan subalgebra.

 su33 > GetComponents(H, CSA);
 $\left[\frac{{1}}{{2}}{,}{-}\frac{{1}}{{2}}{,}{0}{,}{-}\frac{{1}}{{2}}{}{I}{,}\frac{{1}}{{2}}{}{I}\right]$ (2.7)

Here are the root spaces for and

 su33 > X := RootSpace(alpha, RSD);
 ${X}{:=}{\mathrm{E12}}{+}{I}{}{\mathrm{Ei12}}$ (2.8)
 su33 > Y := RootSpace(-alpha, RSD);
 ${Y}{:=}{\mathrm{E21}}{+}{I}{}{\mathrm{Ei21}}$ (2.9)

We check that defines a Lie subalgebra.

 $\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{2}{}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{-}{2}{}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{4}{}{\mathrm{e1}}\right]$ (2.10)

If we scale the vectors X and Y then the structure equations take the standard form for $\mathrm{sl}\left(2\right)$.

 $\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{2}{}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{-}{2}{}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}\right]$ (2.11)

Example 2.

We illustrate how to use RootToCartanSubalgebraElementH(RSD) to calculate the Cartan matrix for We first calculate the for the simple roots $\mathrm{α}$.

 su33 > SR := P["SimpleRoots"];
 ${\mathrm{SR}}{:=}\left[\left[\begin{array}{c}{1}\\ {-}{1}\\ {0}\\ {I}\\ {-}{I}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {1}\\ {-}{1}\\ {I}\\ {2}{}{I}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {2}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {1}\\ {-}{1}\\ {-}{I}\\ {-}{2}{}{I}\end{array}\right]{,}\left[\begin{array}{c}{1}\\ {-}{1}\\ {0}\\ {-}{I}\\ {I}\end{array}\right]\right]$ (2.12)
 su33 > Halpha := map(RootToCartanSubalgebraElementH, SR, RSD);
 ${\mathrm{Halpha}}{:=}\left[{-}\frac{{I}}{{2}}{}{\mathrm{Ei11}}{+}\frac{{I}}{{2}}{}{\mathrm{Ei22}}{+}\frac{{1}}{{2}}{}{\mathrm{E11}}{-}\frac{{1}}{{2}}{}{\mathrm{E22}}{,}{-}\frac{{I}}{{2}}{}{\mathrm{Ei22}}{+}\frac{{1}}{{2}}{}{\mathrm{E22}}{-}\frac{{1}}{{2}}{}{\mathrm{E33}}{,}{\mathrm{E33}}{,}\frac{{I}}{{2}}{}{\mathrm{Ei22}}{+}\frac{{1}}{{2}}{}{\mathrm{E22}}{-}\frac{{1}}{{2}}{}{\mathrm{E33}}{,}\frac{{I}}{{2}}{}{\mathrm{Ei11}}{-}\frac{{I}}{{2}}{}{\mathrm{Ei22}}{+}\frac{{1}}{{2}}{}{\mathrm{E11}}{-}\frac{{1}}{{2}}{}{\mathrm{E22}}\right]$ (2.13)

Then we calculate the Killing form , restricted to subspace [

 su33 > B := Killing(Halpha);
 ${B}{:=}\left[\begin{array}{rrrrr}{24}& {-}{12}& {0}& {0}& {0}\\ {-}{12}& {24}& {-}{12}& {0}& {0}\\ {0}& {-}{12}& {24}& {-}{12}& {0}\\ {0}& {0}& {-}{12}& {24}& {-}{12}\\ {0}& {0}& {0}& {-}{12}& {24}\end{array}\right]$ (2.14)

The Cartan matrix is given by normalizing the entries of $B.$

 su33 > C := Matrix(5, 5, (i, j) -> 2*B[i ,j]/B[i, i]);
 ${C}{:=}\left[\begin{array}{rrrrr}{2}& {-}{1}& {0}& {0}& {0}\\ {-}{1}& {2}& {-}{1}& {0}& {0}\\ {0}& {-}{1}& {2}& {-}{1}& {0}\\ {0}& {0}& {-}{1}& {2}& {-}{1}\\ {0}& {0}& {0}& {-}{1}& {2}\end{array}\right]$ (2.15)

The Lie algebra is a rank 5 simple Lie algebra of type "A". The matrix in (2.15) is therefore correct.

 su33 > CartanMatrix("A", 5);
 $\left[\begin{array}{rrrrr}{2}& {-}{1}& {0}& {0}& {0}\\ {-}{1}& {2}& {-}{1}& {0}& {0}\\ {0}& {-}{1}& {2}& {-}{1}& {0}\\ {0}& {0}& {-}{1}& {2}& {-}{1}\\ {0}& {0}& {0}& {-}{1}& {2}\end{array}\right]$ (2.16)