 Gradation - Maple Help

Query[Gradation] - check if a list of subspaces defines a gradation of a Lie algebra

Calling Sequences

Query([g0, g1, ... , gN], "Gradation")

Parameters

g0, g1,   - a list of independent vectors defining subspaces of a Lie algebra

T         - table, specifying the weights of the basis vectors of a Lie algebra $\mathrm{𝔤}$ Description

 • Let $\mathrm{𝔤}$ be a Lie algebra and let  be a collection of subspaces such that (vector space direct sum).  Assign to the vectors in the weight . Then this decomposition of  defines a graduation of $\mathrm{𝔤}$ if where and  if  is not an assigned weight.
 • Query([g0, g1, g2, ... gN], "Gradation") returns true if the subspaces  define a gradation of the Lie algebra $\mathrm{𝔤}$ with default weights .
 • For the second calling sequence T is a table whose indices are the weights and whose entries are the subspaces ${\mathrm{𝔤}}_{i}$ , that is ,.  The command Query(T, "Gradation") returns true if this systems of weights gives a graduation of $\mathrm{𝔤}$.
 • A general construction of gradations for semi-simple Lie algebras is given by GradeSemiSimpleLieAlgebra .
 • The command Query is part of the DifferentialGeometry:-LieAlgebras package.  It can be used in the form Query(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-Query(...). Examples

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

Example 1.

The Lie algebra of 4 x 4 upper triangular matrices is a 10 dimensional Lie algebra which is naturally graded - consists of the matrices with only non-zero elements on the diagonal, consists of the matrices with non-elements immediately above the diagonal (the super diagonal) and so on.

We use Query to verify this.  First we use the program MatrixAlgebras  to generate the Lie algebra data structure for the Lie algebra of upper triangular matrices.  Here $\mathrm{eij}$ denotes the matrix with a 1 in the $i$-th row and $j$-th column.

 > $\mathrm{L1}≔\mathrm{MatrixAlgebras}\left("Upper",4,\mathrm{alg1}\right)$
 ${\mathrm{L1}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e7}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e7}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e7}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e9}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e9}}\right]{,}\left[{\mathrm{e11}}{,}{\mathrm{e12}}{,}{\mathrm{e13}}{,}{\mathrm{e14}}{,}{\mathrm{e22}}{,}{\mathrm{e23}}{,}{\mathrm{e24}}{,}{\mathrm{e33}}{,}{\mathrm{e34}}{,}{\mathrm{e44}}\right]{,}\left[{\mathrm{ϵ11}}{,}{\mathrm{ϵ12}}{,}{\mathrm{ϵ13}}{,}{\mathrm{ϵ14}}{,}{\mathrm{ϵ22}}{,}{\mathrm{ϵ23}}{,}{\mathrm{ϵ24}}{,}{\mathrm{ϵ33}}{,}{\mathrm{ϵ34}}{,}{\mathrm{ϵ44}}\right]$ (2.1)
 > $\mathrm{DGsetup}\left(\mathrm{L1}\right):$

To display the Lie algebra multiplication table, we need to increase the value of the interface parameter rtablesize.

 alg1 > $\mathrm{interface}\left(\mathrm{rtablesize}=12\right)$
 ${10}$ (2.2)
 alg1 > $\mathrm{MultiplicationTable}\left("LieTable"\right)$
 $\left[\begin{array}{cccccccccccc}{}& {|}& {\mathrm{e11}}& {\mathrm{e12}}& {\mathrm{e13}}& {\mathrm{e14}}& {\mathrm{e22}}& {\mathrm{e23}}& {\mathrm{e24}}& {\mathrm{e33}}& {\mathrm{e34}}& {\mathrm{e44}}\\ {}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}\\ {\mathrm{e11}}& {|}& {0}& {\mathrm{e12}}& {\mathrm{e13}}& {\mathrm{e14}}& {0}& {0}& {0}& {0}& {0}& {0}\\ {\mathrm{e12}}& {|}& {-}{\mathrm{e12}}& {0}& {0}& {0}& {\mathrm{e12}}& {\mathrm{e13}}& {\mathrm{e14}}& {0}& {0}& {0}\\ {\mathrm{e13}}& {|}& {-}{\mathrm{e13}}& {0}& {0}& {0}& {0}& {0}& {0}& {\mathrm{e13}}& {\mathrm{e14}}& {0}\\ {\mathrm{e14}}& {|}& {-}{\mathrm{e14}}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {\mathrm{e14}}\\ {\mathrm{e22}}& {|}& {0}& {-}{\mathrm{e12}}& {0}& {0}& {0}& {\mathrm{e23}}& {\mathrm{e24}}& {0}& {0}& {0}\\ {\mathrm{e23}}& {|}& {0}& {-}{\mathrm{e13}}& {0}& {0}& {-}{\mathrm{e23}}& {0}& {0}& {\mathrm{e23}}& {\mathrm{e24}}& {0}\\ {\mathrm{e24}}& {|}& {0}& {-}{\mathrm{e14}}& {0}& {0}& {-}{\mathrm{e24}}& {0}& {0}& {0}& {0}& {\mathrm{e24}}\\ {\mathrm{e33}}& {|}& {0}& {0}& {-}{\mathrm{e13}}& {0}& {0}& {-}{\mathrm{e23}}& {0}& {0}& {\mathrm{e34}}& {0}\\ {\mathrm{e34}}& {|}& {0}& {0}& {-}{\mathrm{e14}}& {0}& {0}& {-}{\mathrm{e24}}& {0}& {-}{\mathrm{e34}}& {0}& {\mathrm{e34}}\\ {\mathrm{e44}}& {|}& {0}& {0}& {0}& {-}{\mathrm{e14}}& {0}& {0}& {-}{\mathrm{e24}}& {0}& {-}{\mathrm{e34}}& {0}\end{array}\right]$ (2.3)

Now define the 4 subspaces which will define our gradation.

 alg1 > $\mathrm{g0}≔\left[\mathrm{e11},\mathrm{e22},\mathrm{e33},\mathrm{e44}\right]:$$\mathrm{g1}≔\left[\mathrm{e12},\mathrm{e23},\mathrm{e34}\right]:$$\mathrm{g2}≔\left[\mathrm{e13},\mathrm{e24}\right]:$$\mathrm{g3}≔\left[\mathrm{e14}\right]:$
 alg1 > $\mathrm{Query}\left(\left[\mathrm{g0},\mathrm{g1},\mathrm{g2},\mathrm{g3}\right],"Gradation"\right)$
 ${\mathrm{true}}$ (2.4)

Here is the same calculation, obtained using the second calling sequence. First create a table with specifies the weight of each basis vector.

 alg1 > $\mathrm{Gr1}≔\mathrm{table}\left(\left[0=\mathrm{g0},1=\mathrm{g1},2=\mathrm{g2},3=\mathrm{g3}\right]\right)$
 ${\mathrm{Gr}}{:=}{\mathrm{table}}\left(\left[{0}{=}\left[{\mathrm{e11}}{,}{\mathrm{e22}}{,}{\mathrm{e33}}{,}{\mathrm{e44}}\right]{,}{1}{=}\left[{\mathrm{e12}}{,}{\mathrm{e23}}{,}{\mathrm{e34}}\right]{,}{2}{=}\left[{\mathrm{e13}}{,}{\mathrm{e24}}\right]{,}{3}{=}\left[{\mathrm{e14}}\right]\right]\right)$ (2.5)
 alg1 > $\mathrm{Query}\left(\mathrm{Gr1},"Gradation"\right)$
 ${\mathrm{true}}$ (2.6)

Example 2.

In this example we construct the standard gradation for , the Lie algebra of trace-free  matrices. First we use SimpleLieAlgebraData to retrieve the structure equations for this Lie algebra.

 alg1 > $\mathrm{LD2}≔\mathrm{SimpleLieAlgebraData}\left("sl\left(3\right)",\mathrm{alg2},\mathrm{labelformat}="gl",\mathrm{labels}=\left['E','\mathrm{\theta }'\right]\right)$
 ${\mathrm{LD2}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{2}{}{\mathrm{e4}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e7}}\right]{=}{-}{2}{}{\mathrm{e7}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{2}{}{\mathrm{e6}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e8}}\right]{=}{-}{2}{}{\mathrm{e8}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e2}}{+}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e2}}\right]{,}\left[{\mathrm{E11}}{,}{\mathrm{E22}}{,}{\mathrm{E12}}{,}{\mathrm{E13}}{,}{\mathrm{E21}}{,}{\mathrm{E23}}{,}{\mathrm{E31}}{,}{\mathrm{E32}}\right]{,}\left[{\mathrm{θ11}}{,}{\mathrm{θ22}}{,}{\mathrm{θ12}}{,}{\mathrm{θ13}}{,}{\mathrm{θ21}}{,}{\mathrm{θ23}}{,}{\mathrm{θ31}}{,}{\mathrm{θ32}}\right]$ (2.7)
 alg1 > $\mathrm{DGsetup}\left(\mathrm{LD2}\right)$
 ${\mathrm{Lie algebra: alg2}}$ (2.8)
 alg2 > $\mathrm{Gr2}≔\mathrm{table}\left(\left[0=\left[\mathrm{E11},\mathrm{E22}\right],1=\left[\mathrm{E12},\mathrm{E23}\right],2=\left[\mathrm{E13}\right],-1=\left[\mathrm{E21},\mathrm{E32}\right],-2=\left[\mathrm{E31}\right]\right]\right)$
 ${\mathrm{Gr2}}{:=}{\mathrm{table}}\left(\left[{-}{1}{=}\left[{\mathrm{E21}}{,}{\mathrm{E32}}\right]{,}{0}{=}\left[{\mathrm{E11}}{,}{\mathrm{E22}}\right]{,}{-}{2}{=}\left[{\mathrm{E31}}\right]{,}{1}{=}\left[{\mathrm{E12}}{,}{\mathrm{E23}}\right]{,}{2}{=}\left[{\mathrm{E13}}\right]\right]\right)$ (2.9)
 alg2 > $\mathrm{Query}\left(\mathrm{Gr2},"Gradation"\right)$
 ${\mathrm{true}}$ (2.10)