find the Chevalley basis for a real, split semi-simple Lie algebra - Maple Programming Help

Home : Support : Online Help : Mathematics : DifferentialGeometry : LieAlgebras : DifferentialGeometry/LieAlgebras/ChevalleyBasis

LieAlgebras[ChevalleyBasis] - find the Chevalley basis for a real, split semi-simple Lie algebra

Calling Sequences

ChevalleyBasis(CSA, RSD, PosRts)

Parameters

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

RSD      - a table, specifying the root space decomposition of the Lie algebra with respect to the Cartan subalgebra CSA

PosRts   - a list of vectors, specifying a choice of positive roots for the root space decomposition

option   - the keyword argument Algebratype = [A, r] where A is a string "A", "B", "C", "D", "E", "F", or "G" and r is the rank of the Lie algebra.

Description

 • A Chevalley basis is a special choice of basis for a real, split semi-simple Lie algebra. It is adapted to the root space decomposition. In a Chevalley basis, a Cartan subalgebra, the root space decomposition, the Cartan matrix, the simple roots, and the root pattern can be determined by inspection. The structure constants are all integers.
 • The command ChevalleyBasis(CSA, RSD, PosRts) returns a list of vectors defining a Chevalley basis $\mathrm{ℬ}$ =. The structure equations of this basis are

,

Here ${a}_{\mathrm{ij}}$ is the Cartan matrix for $\mathrm{𝔤}$. The roots for ${x}_{a}$, ${x}_{b}$, and ${x}_{c}$ are and The integer is the largest positive integer such that is not a root. See ChevalleyBasisDetails for the algorithm used to construct this basis.

 • Note that in the Chevalley basis all the structure constants are integers and the transformation  ,  is a Lie algebra automorphism.
 • The Chevalley basis is used by the command SplitAndCompactForms to find the split and compact forms of a general semi-simple Lie algebra.

Examples

 > with(DifferentialGeometry): with(LieAlgebras):

Example 1.

We calculate a Chevalley basis for the rank 2 Lie algebra $\mathrm{so}\left(3,2\right)$. We begin with the basis provided by the command SimpleLieAlgebraData.

 > DGsetup(LD);
 ${\mathrm{Lie algebra: so32}}$ (2.1)

We will use the choices of the Cartan subalgebra, root space decomposition, and positive roots for $\mathrm{so}\left(3,2\right)$ contained in SimpleLieAlgebraProperties. (For Lie algebras not created by the SimpleLieAlgebraData command, use CartanSubalgebra, RootSpaceDecomposition, PositiveRoots.)

 so32 > P := SimpleLieAlgebraProperties(so32):
 so32 > CSA := P["CartanSubalgebra"];
 ${\mathrm{CSA}}{:=}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]$ (2.2)
 so32 > RSD := eval(P["RootSpaceDecomposition"]);
 ${\mathrm{RSD}}{:=}{\mathrm{table}}\left(\left[\left[{-}{1}{,}{0}\right]{=}{\mathrm{e9}}{,}\left[{-}{1}{,}{1}\right]{=}{\mathrm{e3}}{,}\left[{0}{,}{-}{1}\right]{=}{\mathrm{e10}}{,}\left[{1}{,}{0}\right]{=}{\mathrm{e7}}{,}\left[{1}{,}{1}\right]{=}{\mathrm{e5}}{,}\left[{1}{,}{-}{1}\right]{=}{\mathrm{e2}}{,}\left[{0}{,}{1}\right]{=}{\mathrm{e8}}{,}\left[{-}{1}{,}{-}{1}\right]{=}{\mathrm{e6}}\right]\right)$ (2.3)
 so32 > PosRts := P["PositiveRoots"];
 ${\mathrm{PosRts}}{:=}\left[\left[\begin{array}{r}{1}\\ {-}{1}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\end{array}\right]\right]$ (2.4)

The Chevalley basis for $\mathrm{so}\left(3,2\right)$ determined by this Cartan subalgebra and choice of positive roots is:

 so32 > CB := ChevalleyBasis(CSA, RSD, PosRts);
 ${\mathrm{CB}}{:=}\left[{\mathrm{e1}}{-}{\mathrm{e4}}{,}{2}{}{\mathrm{e4}}{,}{\mathrm{e2}}{,}{-}{2}{}{\mathrm{e8}}{,}{-}{2}{}{\mathrm{e7}}{,}{-}{2}{}{\mathrm{e5}}{,}{-}{\mathrm{e3}}{,}{-}{\mathrm{e10}}{,}{-}{\mathrm{e9}}{,}{-}\frac{{1}}{{2}}{}{\mathrm{e6}}\right]$ (2.5)

We calculate the structure equations for $\mathrm{so}\left(3,2\right)$ in the Chevalley basis and initialize the Lie algebra in this new basis.

 so32 > newLD := LieAlgebraData(CB, so32CB);
 ${\mathrm{newLD}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{2}{}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e7}}\right]{=}{-}{2}{}{\mathrm{e7}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e8}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e9}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{-}{2}{}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{2}{}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{2}{}{\mathrm{e6}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e7}}\right]{=}{2}{}{\mathrm{e7}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e8}}\right]{=}{-}{2}{}{\mathrm{e8}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e10}}\right]{=}{-}{2}{}{\mathrm{e10}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{2}{}{\mathrm{e6}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e9}}\right]{=}{2}{}{\mathrm{e7}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e9}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e8}}\right]{=}{-}{2}{}{\mathrm{e3}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e9}}\right]{=}{-}{2}{}{\mathrm{e1}}{-}{\mathrm{e2}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e8}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e1}}{-}{\mathrm{e2}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e9}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e9}}\right]{=}{2}{}{\mathrm{e10}}\right]$ (2.6)
 so32 > DGsetup(newLD, '[e1, e2, x1, x2, x3, x4, y1, y2, y3, y4]', '[omega]');
 ${\mathrm{Lie algebra: so32CB}}$ (2.7)

To display the multiplication table for this Lie algebra we use interface to increase the maximum inline array display size.

 so32CB > interface(rtablesize = 15);
 ${10}$ (2.8)
 so32CB > M := MultiplicationTable("LieTable");
 ${M}{:=}\left[\begin{array}{cccccccccccc}{}& {|}& {\mathrm{e1}}& {\mathrm{e2}}& {\mathrm{x1}}& {\mathrm{x2}}& {\mathrm{x3}}& {\mathrm{x4}}& {\mathrm{y1}}& {\mathrm{y2}}& {\mathrm{y3}}& {\mathrm{y4}}\\ {}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}\\ {\mathrm{e1}}& {|}& {0}& {0}& {2}{}{\mathrm{x1}}& {-}{\mathrm{x2}}& {\mathrm{x3}}& {0}& {-}{2}{}{\mathrm{y1}}& {\mathrm{y2}}& {-}{\mathrm{y3}}& {0}\\ {\mathrm{e2}}& {|}& {0}& {0}& {-}{2}{}{\mathrm{x1}}& {2}{}{\mathrm{x2}}& {0}& {2}{}{\mathrm{x4}}& {2}{}{\mathrm{y1}}& {-}{2}{}{\mathrm{y2}}& {0}& {-}{2}{}{\mathrm{y4}}\\ {\mathrm{x1}}& {|}& {-}{2}{}{\mathrm{x1}}& {2}{}{\mathrm{x1}}& {0}& {\mathrm{x3}}& {0}& {0}& {-}{\mathrm{e1}}& {0}& {-}{\mathrm{y2}}& {0}\\ {\mathrm{x2}}& {|}& {\mathrm{x2}}& {-}{2}{}{\mathrm{x2}}& {-}{\mathrm{x3}}& {0}& {2}{}{\mathrm{x4}}& {0}& {0}& {-}{\mathrm{e2}}& {2}{}{\mathrm{y1}}& {-}{\mathrm{y3}}\\ {\mathrm{x3}}& {|}& {-}{\mathrm{x3}}& {0}& {0}& {-}{2}{}{\mathrm{x4}}& {0}& {0}& {\mathrm{x2}}& {-}{2}{}{\mathrm{x1}}& {-}{2}{}{\mathrm{e1}}{-}{\mathrm{e2}}& {\mathrm{y2}}\\ {\mathrm{x4}}& {|}& {0}& {-}{2}{}{\mathrm{x4}}& {0}& {0}& {0}& {0}& {0}& {\mathrm{x3}}& {-}{\mathrm{x2}}& {-}{\mathrm{e1}}{-}{\mathrm{e2}}\\ {\mathrm{y1}}& {|}& {2}{}{\mathrm{y1}}& {-}{2}{}{\mathrm{y1}}& {\mathrm{e1}}& {0}& {-}{\mathrm{x2}}& {0}& {0}& {\mathrm{y3}}& {0}& {0}\\ {\mathrm{y2}}& {|}& {-}{\mathrm{y2}}& {2}{}{\mathrm{y2}}& {0}& {\mathrm{e2}}& {2}{}{\mathrm{x1}}& {-}{\mathrm{x3}}& {-}{\mathrm{y3}}& {0}& {2}{}{\mathrm{y4}}& {0}\\ {\mathrm{y3}}& {|}& {\mathrm{y3}}& {0}& {\mathrm{y2}}& {-}{2}{}{\mathrm{y1}}& {2}{}{\mathrm{e1}}{+}{\mathrm{e2}}& {\mathrm{x2}}& {0}& {-}{2}{}{\mathrm{y4}}& {0}& {0}\\ {\mathrm{y4}}& {|}& {0}& {2}{}{\mathrm{y4}}& {0}& {\mathrm{y3}}& {-}{\mathrm{y2}}& {\mathrm{e1}}{+}{\mathrm{e2}}& {0}& {0}& {0}& {0}\end{array}\right]$ (2.9)

Let us focus in on various parts of the multiplication table. From the first two rows

 so32CB > LinearAlgebra:-SubMatrix(M, [1..4], [1 ..-1]);
 $\left[\begin{array}{cccccccccccc}{}& {|}& {\mathrm{e1}}& {\mathrm{e2}}& {\mathrm{x1}}& {\mathrm{x2}}& {\mathrm{x3}}& {\mathrm{x4}}& {\mathrm{y1}}& {\mathrm{y2}}& {\mathrm{y3}}& {\mathrm{y4}}\\ {}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}\\ {\mathrm{e1}}& {|}& {0}& {0}& {2}{}{\mathrm{x1}}& {-}{\mathrm{x2}}& {\mathrm{x3}}& {0}& {-}{2}{}{\mathrm{y1}}& {\mathrm{y2}}& {-}{\mathrm{y3}}& {0}\\ {\mathrm{e2}}& {|}& {0}& {0}& {-}{2}{}{\mathrm{x1}}& {2}{}{\mathrm{x2}}& {0}& {2}{}{\mathrm{x4}}& {2}{}{\mathrm{y1}}& {-}{2}{}{\mathrm{y2}}& {0}& {-}{2}{}{\mathrm{y4}}\end{array}\right]$ (2.10)

it is clear that  act diagonally and so form a Cartan subalgebra. From the 3rd and 4th columns we can read off the Cartan matrix for $\mathrm{so}\left(3,2\right)$ as the coefficients of:

 so32CB > LinearAlgebra:-SubMatrix(M, [1..4], [1 .. 2, 5..6]);
 $\left[\begin{array}{cccc}{}& {|}& {\mathrm{x1}}& {\mathrm{x2}}\\ {}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}\\ {\mathrm{e1}}& {|}& {2}{}{\mathrm{x1}}& {-}{\mathrm{x2}}\\ {\mathrm{e2}}& {|}& {-}{2}{}{\mathrm{x1}}& {2}{}{\mathrm{x2}}\end{array}\right]$ (2.11)



The vectors  correspond to the roots  with  being the simple roots. Therefore, from the multiplication table

 so32CB > LinearAlgebra:-SubMatrix(M, [1..2, 5..8], [1 .. 2, 5..8]);
 $\left[\begin{array}{cccccc}{}& {|}& {\mathrm{x1}}& {\mathrm{x2}}& {\mathrm{x3}}& {\mathrm{x4}}\\ {}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}\\ {\mathrm{x1}}& {|}& {0}& {\mathrm{x3}}& {0}& {0}\\ {\mathrm{x2}}& {|}& {-}{\mathrm{x3}}& {0}& {2}{}{\mathrm{x4}}& {0}\\ {\mathrm{x3}}& {|}& {0}& {-}{2}{}{\mathrm{x4}}& {0}& {0}\\ {\mathrm{x4}}& {|}& {0}& {0}& {0}& {0}\end{array}\right]$ (2.12)

we can read off the root pattern as , . Finally we note that the vectors  satisfy the same structure equations as .

 so32CB > LinearAlgebra:-SubMatrix(M, [1..2, 9..12], [1 .. 2, 9..12]);
 $\left[\begin{array}{cccccc}{}& {|}& {\mathrm{y1}}& {\mathrm{y2}}& {\mathrm{y3}}& {\mathrm{y4}}\\ {}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}\\ {\mathrm{y1}}& {|}& {0}& {\mathrm{y3}}& {0}& {0}\\ {\mathrm{y2}}& {|}& {-}{\mathrm{y3}}& {0}& {2}{}{\mathrm{y4}}& {0}\\ {\mathrm{y3}}& {|}& {0}& {-}{2}{}{\mathrm{y4}}& {0}& {0}\\ {\mathrm{y4}}& {|}& {0}& {0}& {0}& {0}\end{array}\right]$ (2.13)