 ChevalleyBasis - Maple Help

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

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

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.

 > $\mathrm{LD}â‰”\mathrm{SimpleLieAlgebraData}\left("so\left(3,2\right)",\mathrm{so32}\right):$
 > $\mathrm{DGsetup}\left(\mathrm{LD}\right)$
 ${\mathrm{Lie algebra: so32}}$ (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 > $Pâ‰”\mathrm{SimpleLieAlgebraProperties}\left(\mathrm{so32}\right):$
 so32 > $\mathrm{CSA}â‰”{P}_{"CartanSubalgebra"}$
 ${\mathrm{CSA}}{:=}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]$ (2.2)
 so32 > $\mathrm{RSD}â‰”\mathrm{eval}\left({P}_{"RootSpaceDecomposition"}\right)$
 ${\mathrm{RSD}}{:=}{\mathrm{table}}\left(\left[\left[{-}{1}{,}{0}\right]{=}{\mathrm{e9}}{,}\left[{-}{1}{,}{1}\right]{=}{\mathrm{e3}}{,}\left[{0}{,}{-}{1}\right]{=}{\mathrm{e10}}{,}\left[{1}{,}{0}\right]{=}{\mathrm{e7}}{,}\left[{1}{,}{1}\right]{=}{\mathrm{e5}}{,}\left[{1}{,}{-}{1}\right]{=}{\mathrm{e2}}{,}\left[{0}{,}{1}\right]{=}{\mathrm{e8}}{,}\left[{-}{1}{,}{-}{1}\right]{=}{\mathrm{e6}}\right]\right)$ (2.3)
 so32 > $\mathrm{PosRts}â‰”{P}_{"PositiveRoots"}$ The Chevalley basis for $\mathrm{so}\left(3,2\right)$ determined by this Cartan subalgebra and choice of positive roots is:

 so32 > $\mathrm{CB}â‰”\mathrm{ChevalleyBasis}\left(\mathrm{CSA},\mathrm{RSD},\mathrm{PosRts}\right)$
 ${\mathrm{CB}}{:=}\left[{\mathrm{e1}}{-}{\mathrm{e4}}{,}{2}{}{\mathrm{e4}}{,}{\mathrm{e2}}{,}{-}{2}{}{\mathrm{e8}}{,}{-}{2}{}{\mathrm{e7}}{,}{-}{2}{}{\mathrm{e5}}{,}{-}{\mathrm{e3}}{,}{-}{\mathrm{e10}}{,}{-}{\mathrm{e9}}{,}{-}\frac{{1}}{{2}}{}{\mathrm{e6}}\right]$ (2.4)

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 > $\mathrm{newLD}â‰”\mathrm{LieAlgebraData}\left(\mathrm{CB},\mathrm{so32CB}\right)$
 ${\mathrm{newLD}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{2}{}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e7}}\right]{=}{-}{2}{}{\mathrm{e7}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e8}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e9}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{-}{2}{}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{2}{}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{2}{}{\mathrm{e6}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e7}}\right]{=}{2}{}{\mathrm{e7}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e8}}\right]{=}{-}{2}{}{\mathrm{e8}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e10}}\right]{=}{-}{2}{}{\mathrm{e10}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{2}{}{\mathrm{e6}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e9}}\right]{=}{2}{}{\mathrm{e7}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e9}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e8}}\right]{=}{-}{2}{}{\mathrm{e3}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e9}}\right]{=}{-}{2}{}{\mathrm{e1}}{-}{\mathrm{e2}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e8}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e1}}{-}{\mathrm{e2}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e9}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e9}}\right]{=}{2}{}{\mathrm{e10}}\right]$ (2.5)
 so32 > $\mathrm{DGsetup}\left(\mathrm{newLD},'\left[\mathrm{e1},\mathrm{e2},\mathrm{x1},\mathrm{x2},\mathrm{x3},\mathrm{x4},\mathrm{y1},\mathrm{y2},\mathrm{y3},\mathrm{y4}\right]','\left[\mathrm{ω}\right]'\right)$
 ${\mathrm{Lie algebra: so32CB}}$ (2.6)

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

 so32CB > $\mathrm{interface}\left(\mathrm{rtablesize}=15\right)$
 ${10}$ (2.7)
 so32CB > $Mâ‰”\mathrm{MultiplicationTable}\left("LieTable"\right)$ Let us focus in on various parts of the multiplication table. From the first two rows

 so32CB > $\mathrm{LinearAlgebra}:-\mathrm{SubMatrix}\left(M,\left[1..4\right],\left[1..-1\right]\right)$ 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 > $\mathrm{LinearAlgebra}:-\mathrm{SubMatrix}\left(M,\left[1..4\right],\left[1..2,5..6\right]\right)$ The vectors  correspond to the roots  with  being the simple roots. Therefore, from the multiplication table

 so32CB > $\mathrm{LinearAlgebra}:-\mathrm{SubMatrix}\left(M,\left[1..2,5..8\right],\left[1..2,5..8\right]\right)$ we can read off the root pattern as , . Finally we note that the vectors  satisfy the same structure equations as .

 so32CB > $\mathrm{LinearAlgebra}:-\mathrm{SubMatrix}\left(M,\left[1..2,9..12\right],\left[1..2,9..12\right]\right)$ 