Calling Sequence

Parameters

AlgName           - a name or string, the name to be assigned to the new structure equations for the Lie algebra

SubAlgComponents  - (optional) a string, one of "positive", "negative", "nonnegative", "nonpositive" or a list of integers

SubAlgName        - (optional) a name or a string, the name to be assigned to the Lie subalgebra defined by SubAlgComponents

options           - (optional) keyword arguments order = increasing or order = decreasing, and output = "basis"

Description

 • If is a graded Lie algebra, then it is often desirable to construct a basis for the Lie algebra adapted to the grading. For example, if with dim = 2, dim = 3, dimthen is an adapted basis, in increasing ordering of weights, if  and . The adapted basis is in decreasing order of weights if  and .
 • The calling sequence LieAlgebraData(Grading, AlgName) returns the structure equations for the adapted basis (in increasing order) for the given grading. These structure equations can be passed to DGsetup to initialize the original Lie algebra in the new adapted basis.
 • For any graded Lie algebra, the negative, non-positive, positive, non-positive components constitute a Lie subalgebra which is frequently needed. The second calling sequence LieAlgebraData(Grading, AlgName, SubAlgComponents, SubAlgName) returns 2 sets of structure equations, the first are the structure equations for the adapted basis (in increasing order) for the given grading, and the second are the structure equations for the subalgebra specified by the third argument.
 • With the keyword argument output = "basis", both the structure equations and the adapted basis are returned.

Examples

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

Example 1.

We illustrate the first calling sequence for LieAlgebraData using the Lie algebra and a 5-step gradation (Grading of semi-simple Lie algebras are easily constructed with GradedSemiSimpleLieAlgebra ). Use SimpleLieAlgebraData and DGsetup to initialize $\mathrm{sl}\left(3\right)$.

 > $\mathrm{LD}≔\mathrm{SimpleLieAlgebraData}\left("sl\left(3\right)",\mathrm{sl3}\right)$
 ${\mathrm{LD}}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{2}{}{\mathrm{e4}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e7}}\right]{=}{-}{2}{}{\mathrm{e7}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{2}{}{\mathrm{e6}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e8}}\right]{=}{-}{2}{}{\mathrm{e8}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e1}}{-}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e2}}\right]$ (2.1)
 > $\mathrm{DGsetup}\left(\mathrm{LD}\right)$
 ${\mathrm{Lie algebra: sl3}}$ (2.2)

Here is the grading we shall use.

 sl3 > $G≔\mathrm{table}\left(\left[0=\left[\mathrm{e1},\mathrm{e2}\right],1=\left[\mathrm{e3},\mathrm{e6}\right],2=\left[\mathrm{e4}\right],-2=\left[\mathrm{e7}\right],-1=\left[\mathrm{e5},\mathrm{e8}\right]\right]\right)$
 ${G}{≔}{table}{}\left(\left[{-1}{=}\left[{\mathrm{e5}}{,}{\mathrm{e8}}\right]{,}{0}{=}\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{,}{-2}{=}\left[{\mathrm{e7}}\right]{,}{1}{=}\left[{\mathrm{e3}}{,}{\mathrm{e6}}\right]{,}{2}{=}\left[{\mathrm{e4}}\right]\right]\right)$ (2.3)

Here are the structure equations for $\mathrm{sl}\left(3\right)$ adapted to the basis .

 sl3 > $\mathrm{LD1},\mathrm{B1}≔\mathrm{LieAlgebraData}\left(G,\mathrm{sl3a},\mathrm{output}="basis"\right)$
 ${\mathrm{LD1}}{,}{\mathrm{B1}}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{2}{}{\mathrm{e1}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e5}}{-}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e7}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{2}{}{\mathrm{e3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e7}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e8}}\right]{=}{2}{}{\mathrm{e8}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e7}}\right]{=}{2}{}{\mathrm{e7}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e8}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e8}}\right]{,}\left[{\mathrm{e7}}{,}{\mathrm{e5}}{,}{\mathrm{e8}}{,}{\mathrm{e1}}{,}{\mathrm{e2}}{,}{\mathrm{e3}}{,}{\mathrm{e6}}{,}{\mathrm{e4}}\right]$ (2.4)
 sl3 > $\mathrm{DGsetup}\left(\mathrm{LD1}\right)$
 ${\mathrm{Lie algebra: sl3a}}$ (2.5)

We see that the grading weights are increasing in this new basis.

 sl3a > $\mathrm{Tools}:-\mathrm{DGinfo}\left("table","Grading"\right)$
 ${table}{}\left(\left[{-1}{=}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{,}{0}{=}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{,}{-2}{=}\left[{\mathrm{e1}}\right]{,}{1}{=}\left[{\mathrm{e6}}{,}{\mathrm{e7}}\right]{,}{2}{=}\left[{\mathrm{e8}}\right]\right]\right)$ (2.6)

With the keyword argument order = decreasing we obtain:

 sl3a > $\mathrm{LD2},\mathrm{B2}≔\mathrm{LieAlgebraData}\left(G,\mathrm{sl3b},\mathrm{order}="decreasing",\mathrm{output}="basis"\right)$
 ${\mathrm{LD2}}{,}{\mathrm{B2}}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{-}{2}{}{\mathrm{e1}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e4}}{-}{\mathrm{e5}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{-}{2}{}{\mathrm{e3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e8}}\right]{=}{-}{2}{}{\mathrm{e8}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e7}}\right]{=}{-}{2}{}{\mathrm{e7}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e8}}\right]{,}\left[{\mathrm{e4}}{,}{\mathrm{e3}}{,}{\mathrm{e6}}{,}{\mathrm{e1}}{,}{\mathrm{e2}}{,}{\mathrm{e5}}{,}{\mathrm{e8}}{,}{\mathrm{e7}}\right]$ (2.7)
 newsl3 > $\mathrm{DGsetup}\left(\mathrm{LD2}\right)$
 ${\mathrm{Lie algebra: sl3b}}$ (2.8)

Now the grading weights are decreasing in the new basis.

 sl3b > $\mathrm{Tools}:-\mathrm{DGinfo}\left("table","Grading"\right)$
 ${table}{}\left(\left[{-1}{=}\left[{\mathrm{e6}}{,}{\mathrm{e7}}\right]{,}{0}{=}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{,}{-2}{=}\left[{\mathrm{e8}}\right]{,}{1}{=}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{,}{2}{=}\left[{\mathrm{e1}}\right]\right]\right)$ (2.9)

With the second calling sequence we can, for example, also obtain the structure equations for just the negatively graded components of.

 sl3 > $\mathrm{LD3},\mathrm{LD4},\mathrm{B3},\mathrm{B4}≔\mathrm{LieAlgebraData}\left(G,\mathrm{sl3c},"negative",\mathrm{sl3minus},\mathrm{output}="basis"\right)$
 ${\mathrm{LD3}}{,}{\mathrm{LD4}}{,}{\mathrm{B3}}{,}{\mathrm{B4}}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{2}{}{\mathrm{e1}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e5}}{-}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e7}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{2}{}{\mathrm{e3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e7}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e8}}\right]{=}{2}{}{\mathrm{e8}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e7}}\right]{=}{2}{}{\mathrm{e7}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e8}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e8}}\right]{,}\left[\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e1}}\right]{,}\left[{\mathrm{e7}}{,}{\mathrm{e5}}{,}{\mathrm{e8}}{,}{\mathrm{e1}}{,}{\mathrm{e2}}{,}{\mathrm{e3}}{,}{\mathrm{e6}}{,}{\mathrm{e4}}\right]{,}\left[{\mathrm{e7}}{,}{\mathrm{e5}}{,}{\mathrm{e8}}\right]$ (2.10)

We see that the negatively graded components give the subalgebra:

 sl3 > $\mathrm{B4}$
 $\left[{\mathrm{e7}}{,}{\mathrm{e5}}{,}{\mathrm{e8}}\right]$ (2.11)

with structure equations:

 sl3 > $\mathrm{LD4}$
 $\left[\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e1}}\right]$ (2.12)
 sl3 >