find the coroot of a root vector for a semi-simple Lie algebra - Maple Programming Help

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LieAlgebras[CoRoot] - find the coroot of a root vector for a semi-simple Lie algebra

Calling Sequences

CoRoot(${\mathbf{α}}$, CSA, option)

Parameters

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

CSA      - a list of $r$ vectors in a Lie algebra, defining a Cartan subalgebra

option   - an  non-singular matrix, defining the restriction of the Killing form to the Cartan subalgebra

Description

 • Let $\mathrm{𝔤}$ be a semi-simple Lie algebra, a Cartan subalgebra, and the associated set of roots. Let be the Killing form. If then the coroot of $\mathrm{α}$ is the unique vector such that. Let  be a basis for and with inverse. Then , where .
 • The calling sequence CoRoot(${\mathbf{α}}$, CSA) returns the vector ${T}_{\mathrm{α}}$.
 • In a situation involving repeated calls to CoRoot, efficiency can be dramatically improved by using the optional 3rd argument to specify the restriction of the Killing form.

Examples

 > with(DifferentialGeometry): with(LieAlgebras):

Example 1.

We use the command SimpleLieAlgebraData to retrieve the structure equations for the rank 3 Lie algebra we initialize this algebra, and we calculate the coroots of several root vectors.

 ${\mathrm{LD}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e6}}\right]{=}{2}{}{\mathrm{e6}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e9}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e10}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e12}}\right]{=}{\mathrm{e12}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e13}}\right]{=}{-}{2}{}{\mathrm{e13}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e14}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e15}}\right]{=}{-}{\mathrm{e15}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e7}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e8}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e9}}\right]{=}{2}{}{\mathrm{e9}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e11}}\right]{=}{-}{\mathrm{e11}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e12}}\right]{=}{\mathrm{e12}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e13}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e14}}\right]{=}{-}{2}{}{\mathrm{e14}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e15}}\right]{=}{-}{\mathrm{e15}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e9}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e10}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e11}}\right]{=}{\mathrm{e11}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e12}}\right]{=}{2}{}{\mathrm{e12}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e13}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e14}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e15}}\right]{=}{-}{2}{}{\mathrm{e15}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e1}}{-}{\mathrm{e2}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e11}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e14}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e1}}{-}{\mathrm{e3}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e11}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e12}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e15}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e9}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e12}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e13}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e14}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e15}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e11}}\right]{=}{-}{\mathrm{e10}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e13}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e7}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e11}}\right]{=}{\mathrm{e2}}{-}{\mathrm{e3}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e12}}\right]{=}{\mathrm{e9}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e15}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e11}}\right]{=}{-}{\mathrm{e12}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e13}}\right]{=}{\mathrm{e7}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e14}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e15}}\right]{=}{\mathrm{e8}}{,}\left[{\mathrm{e10}}{,}{\mathrm{e15}}\right]{=}{-}{\mathrm{e13}}{,}\left[{\mathrm{e11}}{,}{\mathrm{e15}}\right]{=}{-}{\mathrm{e14}}{,}\left[{\mathrm{e12}}{,}{\mathrm{e13}}\right]{=}{\mathrm{e10}}{,}\left[{\mathrm{e12}}{,}{\mathrm{e14}}\right]{=}{\mathrm{e11}}{,}\left[{\mathrm{e12}}{,}{\mathrm{e15}}\right]{=}{\mathrm{e3}}\right]$ (2.1)
 > DGsetup(LD);
 ${\mathrm{Lie algebra: sl4}}$ (2.2)

We obtain the Cartan subalgebra and the positive roots using SimpleLieAlgebraProperties

 M > P:= SimpleLieAlgebraProperties(sl4):
 sl4 > CSA := P["CartanSubalgebra"];
 ${\mathrm{CSA}}{:=}\left[{\mathrm{e1}}{,}{\mathrm{e2}}{,}{\mathrm{e3}}\right]$ (2.3)
 sl4 > Delta := P["PositiveRoots"];
 ${\mathrm{Δ}}{:=}\left[\left[\begin{array}{r}{1}\\ {-}{1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {-}{1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {2}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {0}\\ {-}{1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {2}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{2}\\ {1}\\ {1}\end{array}\right]\right]$ (2.4)

Calculate the coroot for the first root ${\mathrm{Δ}}_{1}$.

 sl4 > alpha := Delta[1];
 ${\mathrm{α}}{:=}\left[\begin{array}{r}{1}\\ {-}{1}\\ {0}\end{array}\right]$ (2.5)
 sl4 > CoRoot(alpha, CSA);
 $\frac{{1}}{{8}}{}{\mathrm{e1}}{-}\frac{{1}}{{8}}{}{\mathrm{e2}}$ (2.6)

Calculate the coroot for the last root ${\mathrm{Δ}}_{6}$.

 sl4 > beta := Delta[-1];
 ${\mathrm{β}}{:=}\left[\begin{array}{r}{2}\\ {1}\\ {1}\end{array}\right]$ (2.7)
 sl4 > CoRoot(beta, CSA);
 $\frac{{1}}{{8}}{}{\mathrm{e1}}$ (2.8)

Example 2.

We repeat the calculation the first coroot from Example 1 using the optional calling sequence. The restriction of the Killing form to the Cartan subalgebra is needed.

 sl4 > B := Killing([e1, e2, e3]);
 ${B}{:=}\left[\begin{array}{rrr}{16}& {8}& {8}\\ {8}& {16}& {8}\\ {8}& {8}& {16}\end{array}\right]$ (2.9)
 sl4 > CoRoot(alpha, CSA, B);
 $\frac{{1}}{{8}}{}{\mathrm{e1}}{-}\frac{{1}}{{8}}{}{\mathrm{e2}}$ (2.10)

Here is the same computation in components.

 sl4 > T := B^(-1).alpha;
 ${T}{:=}\left[\begin{array}{c}\frac{{1}}{{8}}\\ {-}\frac{{1}}{{8}}\\ {0}\end{array}\right]$ (2.11)