 RootToCartanSubalgebraElementH - Maple 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

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

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.

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

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

 su33 > $P≔\mathrm{SimpleLieAlgebraProperties}\left(\mathrm{su33}\right):$

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

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

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

 su33 > $\mathrm{RSD}≔\mathrm{eval}\left({P}_{"RootSpaceDecomposition"}\right)$
 ${\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 > $\mathrm{PR}≔{P}_{"PositiveRoots"}$ Let us find ${H}_{{\mathrm{α}}_{}},$where ${\mathrm{α}}_{}$ is the first root

 su33 > $\mathrm{α}≔{\mathrm{PR}}_{1}$ su33 > $H≔\mathrm{RootToCartanSubalgebraElementH}\left(\mathrm{α},\mathrm{RSD}\right)$
 ${H}{:=}{-}\frac{{I}}{{2}}{}{\mathrm{Ei11}}{+}\frac{{I}}{{2}}{}{\mathrm{Ei22}}{+}\frac{{1}}{{2}}{}{\mathrm{E11}}{-}\frac{{1}}{{2}}{}{\mathrm{E22}}$ (2.4)

We check that is in the Cartan subalgebra.

 su33 > $\mathrm{GetComponents}\left(H,\mathrm{CSA}\right)$
 $\left[\frac{{1}}{{2}}{,}{-}\frac{{1}}{{2}}{,}{0}{,}{-}\frac{{1}}{{2}}{}{I}{,}\frac{{1}}{{2}}{}{I}\right]$ (2.5)

Here are the root spaces for and

 su33 > $X≔\mathrm{RootSpace}\left(\mathrm{α},\mathrm{RSD}\right)$
 ${X}{:=}{\mathrm{E12}}{+}{I}{}{\mathrm{Ei12}}$ (2.6)
 su33 > $Y≔\mathrm{RootSpace}\left(-\mathrm{α},\mathrm{RSD}\right)$
 ${Y}{:=}{\mathrm{E21}}{+}{I}{}{\mathrm{Ei21}}$ (2.7)

We check that defines a Lie subalgebra.

 su33 > $\mathrm{LieAlgebraData}\left(\left[H,X,Y\right]\right)$
 $\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{2}{}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{-}{2}{}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{4}{}{\mathrm{e1}}\right]$ (2.8)

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

 su33 > $\mathrm{LieAlgebraData}\left(\left[H,\frac{1X}{2},\frac{1Y}{2}\right]\right)$
 $\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{2}{}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{-}{2}{}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}\right]$ (2.9)

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 > $\mathrm{SR}≔{P}_{"SimpleRoots"}$ su33 > $\mathrm{Halpha}≔\mathrm{map}\left(\mathrm{RootToCartanSubalgebraElementH},\mathrm{SR},\mathrm{RSD}\right)$
 ${\mathrm{Halpha}}{:=}\left[{-}\frac{{I}}{{2}}{}{\mathrm{Ei11}}{+}\frac{{I}}{{2}}{}{\mathrm{Ei22}}{+}\frac{{1}}{{2}}{}{\mathrm{E11}}{-}\frac{{1}}{{2}}{}{\mathrm{E22}}{,}{-}\frac{{I}}{{2}}{}{\mathrm{Ei22}}{+}\frac{{1}}{{2}}{}{\mathrm{E22}}{-}\frac{{1}}{{2}}{}{\mathrm{E33}}{,}{\mathrm{E33}}{,}\frac{{I}}{{2}}{}{\mathrm{Ei22}}{+}\frac{{1}}{{2}}{}{\mathrm{E22}}{-}\frac{{1}}{{2}}{}{\mathrm{E33}}{,}\frac{{I}}{{2}}{}{\mathrm{Ei11}}{-}\frac{{I}}{{2}}{}{\mathrm{Ei22}}{+}\frac{{1}}{{2}}{}{\mathrm{E11}}{-}\frac{{1}}{{2}}{}{\mathrm{E22}}\right]$ (2.10)

Then we calculate the Killing form , restricted to subspace [

 su33 > $B≔\mathrm{Killing}\left(\mathrm{Halpha}\right)$ The Cartan matrix is given by normalizing the entries of $B.$

 su33 > $C≔\mathrm{Matrix}\left(5,5,\left(i,j\right)→\frac{2{B}_{i,j}}{{B}_{i,i}}\right)$ The Lie algebra is a rank 5 simple Lie algebra of type "A". The matrix in  is therefore correct.

 su33 > $\mathrm{CartanMatrix}\left("A",5\right)$ 