find the sequence of roots through a given root of a semi-simple Lie algebra - Maple Programming Help

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LieAlgebras[RootString] - find the sequence of roots through a given root of a semi-simple Lie algebra

Calling Sequences

RootString(${\mathbf{α}}$, ${\mathbf{β}}$, ${\mathbf{Δ}}$, option)

Parameters

$\mathrm{α}$       - a vector, defining a root vector for a semi-simple Lie algebra

$\mathrm{β}$       - a vector, defining a root vector for a semi-simple Lie algebra

$\mathrm{Δ}$       - a list of vectors, defining a list of root vectors for a semi-simple Lie algebra and containing $\mathrm{α}$ and $\mathrm{β}$

option       - the keyword argument output = "stringlengths"

Description

 • Let $\mathrm{𝔤}$ be a semi-simple Lie algebra, $\mathrm{𝔥}$ a Cartan subalgebra, and the associated set of roots. If then the $\mathrm{α}$-string through $\mathrm{β}$ is the maximal sequence of roots of the form

where $p,q$ are non-negative integers.

 • The calling sequence RootString(${\mathbf{α}}$, ${\mathbf{β}}$, ${\mathbf{Δ}}$) returns the $\mathrm{α}$-string of roots through $\mathrm{β}$. The calling sequence RootString(${\mathbf{α}}$, ${\mathbf{β}}$, ${\mathbf{Δ}}$, output = "stringlengths") returns the list of non-negative integers .

Examples

 > with(DifferentialGeometry): with(LieAlgebras):

Example 1.

We initialize the split real form of the exceptional Lie algebra ${g}_{2}$ and retrieve the root space decomposition and the list of all roots. We then calculate some root strings. The structure equations for ${g}_{2}$ are obtained using SimpleLieAlgebraData.

 > LD := SimpleLieAlgebraData("g(2,Split)", g2);
 ${\mathrm{LD}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{2}{}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{-}{3}{}{\mathrm{e4}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e7}}\right]{=}{3}{}{\mathrm{e7}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e9}}\right]{=}{-}{2}{}{\mathrm{e9}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e10}}\right]{=}{3}{}{\mathrm{e10}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e11}}\right]{=}{\mathrm{e11}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e12}}\right]{=}{-}{\mathrm{e12}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e13}}\right]{=}{-}{3}{}{\mathrm{e13}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{2}{}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e8}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e9}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e10}}\right]{=}{-}{2}{}{\mathrm{e10}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e11}}\right]{=}{-}{\mathrm{e11}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e13}}\right]{=}{\mathrm{e13}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e14}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{2}{}{\mathrm{e6}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e6}}\right]{=}{-}{3}{}{\mathrm{e7}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e11}}\right]{=}{-}{3}{}{\mathrm{e10}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e12}}\right]{=}{-}{2}{}{\mathrm{e11}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e13}}\right]{=}{\mathrm{e12}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e11}}\right]{=}{\mathrm{e9}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e14}}\right]{=}{\mathrm{e13}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{-}{3}{}{\mathrm{e8}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e9}}\right]{=}{3}{}{\mathrm{e4}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e11}}\right]{=}{-}{\mathrm{e1}}{-}{3}{}{\mathrm{e2}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e12}}\right]{=}{2}{}{\mathrm{e9}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e14}}\right]{=}{\mathrm{e12}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e9}}\right]{=}{2}{}{\mathrm{e5}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e11}}\right]{=}{-}{2}{}{\mathrm{e3}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e12}}\right]{=}{-}{2}{}{\mathrm{e1}}{-}{3}{}{\mathrm{e2}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e9}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e11}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e12}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e1}}{-}{\mathrm{e2}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e10}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e11}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e12}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e13}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e1}}{-}{2}{}{\mathrm{e2}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e11}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e11}}\right]{=}{2}{}{\mathrm{e12}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e12}}\right]{=}{-}{3}{}{\mathrm{e13}}{,}\left[{\mathrm{e10}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e14}}{,}\left[{\mathrm{e11}}{,}{\mathrm{e12}}\right]{=}{-}{3}{}{\mathrm{e14}}\right]$ (2.1)

Initialize the Lie algebra with DGsetup.

 > DGsetup(LD);
 ${\mathrm{Lie algebra: g2}}$ (2.2)

The root space decomposition is retrieved, without calculation, using SimpleLieAlgebraProperties

 > P := SimpleLieAlgebraProperties(g2):

Here is the root space decomposition and the list of all positive roots.

 > RSD := eval(P["RootSpaceDecomposition"]);
 ${\mathrm{RSD}}{:=}{\mathrm{table}}\left(\left[\left[{-}{1}{,}{1}\right]{=}{\mathrm{e5}}{,}\left[{-}{3}{,}{2}\right]{=}{\mathrm{e4}}{,}\left[{0}{,}{1}\right]{=}{\mathrm{e8}}{,}\left[{3}{,}{-}{1}\right]{=}{\mathrm{e7}}{,}\left[{2}{,}{-}{1}\right]{=}{\mathrm{e3}}{,}\left[{1}{,}{0}\right]{=}{\mathrm{e6}}{,}\left[{0}{,}{-}{1}\right]{=}{\mathrm{e14}}{,}\left[{-}{2}{,}{1}\right]{=}{\mathrm{e9}}{,}\left[{-}{3}{,}{1}\right]{=}{\mathrm{e13}}{,}\left[{3}{,}{-}{2}\right]{=}{\mathrm{e10}}{,}\left[{-}{1}{,}{0}\right]{=}{\mathrm{e12}}{,}\left[{1}{,}{-}{1}\right]{=}{\mathrm{e11}}\right]\right)$ (2.3)
 g2 > Delta := LieAlgebraRoots(RSD);
 ${\mathrm{Δ}}{:=}\left[\left[\begin{array}{r}{-}{1}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{-}{3}\\ {2}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{3}\\ {-}{1}\end{array}\right]{,}\left[\begin{array}{r}{2}\\ {-}{1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {-}{1}\end{array}\right]{,}\left[\begin{array}{r}{-}{2}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{-}{3}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{3}\\ {-}{2}\end{array}\right]{,}\left[\begin{array}{r}{-}{1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {-}{1}\end{array}\right]\right]$ (2.4)

Define two roots and $\mathrm{β}$.

 g2 > alpha := <2, -1>;
 ${\mathrm{α}}{:=}\left[\begin{array}{r}{2}\\ {-}{1}\end{array}\right]$ (2.5)
 g2 > beta := <-3, 2>;
 ${\mathrm{β}}{:=}\left[\begin{array}{r}{-}{3}\\ {2}\end{array}\right]$ (2.6)

Calculate the $\mathrm{α}$-string through $\mathrm{β}.$

 g2 > RootString(alpha, beta, Delta);
 $\left[\left[\begin{array}{r}{-}{3}\\ {2}\end{array}\right]{,}\left[\begin{array}{r}{-}{1}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{3}\\ {-}{1}\end{array}\right]\right]$ (2.7)

With the optional keyword argument output = "stringlengths", we obtain the lengths of the $\mathrm{α}$-string through in the negative and positive directions.

 g2 > RootString(alpha, beta, Delta, output = "stringlengths");
 $\left[{0}{,}{3}\right]$ (2.8)

Thus the $\mathrm{α}$-string through $\mathrm{β}$ in (2.7) is given explicitly by.

 g2 > [beta, beta + alpha, beta +2*alpha, beta + 3*alpha];
 $\left[\left[\begin{array}{r}{-}{3}\\ {2}\end{array}\right]{,}\left[\begin{array}{r}{-}{1}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{3}\\ {-}{1}\end{array}\right]\right]$ (2.9)



Example 2.

Here is another example of a root string for the exceptional Lie algebra ${g}_{2}$.

 g2 > alpha := <2, -1>;
 ${\mathrm{α}}{:=}\left[\begin{array}{r}{2}\\ {-}{1}\end{array}\right]$ (2.10)
 g2 > beta := <1, 0>;
 ${\mathrm{β}}{:=}\left[\begin{array}{r}{1}\\ {0}\end{array}\right]$ (2.11)

The root $\mathrm{α}$-string through $\mathrm{β}$ is now

 g2 > RootString(alpha, beta, Delta);
 $\left[\left[\begin{array}{r}{-}{3}\\ {2}\end{array}\right]{,}\left[\begin{array}{r}{-}{1}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{3}\\ {-}{1}\end{array}\right]\right]$ (2.12)

and the string lengths are

 g2 > RootString(alpha, beta, Delta, output = "stringlengths");
 $\left[{2}{,}{1}\right]$ (2.13)



Thus, the root string (2.12) is explicitly given by

 g2 > [beta -2*alpha, beta - alpha, beta, beta + alpha];
 $\left[\left[\begin{array}{r}{-}{3}\\ {2}\end{array}\right]{,}\left[\begin{array}{r}{-}{1}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{3}\\ {-}{1}\end{array}\right]\right]$ (2.14)

Example 3.

Root strings can also be calculated for abstract roots systems, that is, a set of vectors satisfying the standard axioms of a root system and not explicitly defined from the root space decomposition of a semi-simple Lie algebra. The positive roots of an abstract root systems can be calculated with the PositiveRoots command.

 g2 > AbstractRoots := PositiveRoots("B", 3);
 ${\mathrm{AbstractRoots}}{:=}\left[\left[\begin{array}{r}{1}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {2}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {2}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {2}\\ {2}\end{array}\right]\right]$ (2.15)
 g2 > Delta := [seq(-v, v = AbstractRoots), seq(v, v = AbstractRoots)];
 ${\mathrm{Δ}}{:=}\left[\left[\begin{array}{r}{-}{1}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {-}{1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {-}{1}\end{array}\right]{,}\left[\begin{array}{r}{-}{1}\\ {-}{1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {-}{1}\\ {-}{1}\end{array}\right]{,}\left[\begin{array}{r}{-}{1}\\ {-}{1}\\ {-}{1}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {-}{1}\\ {-}{2}\end{array}\right]{,}\left[\begin{array}{r}{-}{1}\\ {-}{1}\\ {-}{2}\end{array}\right]{,}\left[\begin{array}{r}{-}{1}\\ {-}{2}\\ {-}{2}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {2}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {2}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {2}\\ {2}\end{array}\right]\right]$ (2.16)

Here are are 2 roots and their root string.

 g2 > alpha := Delta[12];
 ${\mathrm{α}}{:=}\left[\begin{array}{r}{0}\\ {0}\\ {1}\end{array}\right]$ (2.17)
 g2 > beta := Delta[11];
 ${\mathrm{β}}{:=}\left[\begin{array}{r}{0}\\ {1}\\ {0}\end{array}\right]$ (2.18)
 g2 > RootString(alpha, beta, Delta);
 $\left[\left[\begin{array}{r}{0}\\ {1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {2}\end{array}\right]\right]$ (2.19)