LieAlgebraData[Grading] - give a new basis and new structure equations for a Lie algebra which are adapted to a grading
Calling Sequence
LieAlgebraData(Grading, AlgName, SubAlgComponents, SubAlgName, options)
Parameters
Grading - a table, specifying a grading of a Lie algebra
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
Examples
If 𝔤 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 𝔤 = 𝔤−1⊕ 𝔤0 ⊕ 𝔤1 with dim 𝔤−1 = 2, dim 𝔤0 = 3, dim 𝔤1 = 1 then e1, e2, e3, e4, e5, e6 is an adapted basis, in increasing ordering of weights, if 𝔤−1= span e1, e2, 𝔤0 = span e3, e4, e5 and 𝔤1 = span e6. The adapted basis is in decreasing order of weights if 𝔤1 = span e1, 𝔤0 = span e2, e3, e4 and 𝔤−1 = span e5,e6.
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.
restart:withDifferentialGeometry:withLieAlgebras:
Example 1.
We illustrate the first calling sequence for LieAlgebraData using the Lie algebra sl3 and a 5-step gradation (Grading of semi-simple Lie algebras are easily constructed with GradedSemiSimpleLieAlgebra ). Use SimpleLieAlgebraData and DGsetup to initialize sl3.
LD≔SimpleLieAlgebraDatasl(3),sl3
LD≔e1,e3=e3,e1,e4=2e4,e1,e5=−e5,e1,e6=e6,e1,e7=−2e7,e1,e8=−e8,e2,e3=−e3,e2,e4=e4,e2,e5=e5,e2,e6=2e6,e2,e7=−e7,e2,e8=−2e8,e3,e5=e1−e2,e3,e6=e4,e3,e7=−e8,e4,e5=−e6,e4,e7=e1,e4,e8=e3,e5,e8=−e7,e6,e7=e5,e6,e8=e2
DGsetupLD
Lie algebra: sl3
Here is the grading we shall use.
G≔table0=e1,e2,1=e3,e6,2=e4,−2=e7,−1=e5,e8
G≔table−1=e5,e8,0=e1,e2,−2=e7,1=e3,e6,2=e4
Here are the structure equations for sl3 adapted to the basis e7, e5,e8,e1, e2, e3, e6, e4.
LD1,B1≔LieAlgebraDataG,sl3a,output=basis
LD1,B1≔e1,e4=2e1,e1,e5=e1,e1,e6=e3,e1,e7=−e2,e1,e8=−e4,e2,e3=−e1,e2,e4=e2,e2,e5=−e2,e2,e6=e5−e4,e2,e8=e7,e3,e4=e3,e3,e5=2e3,e3,e7=−e5,e3,e8=−e6,e4,e6=e6,e4,e7=e7,e4,e8=2e8,e5,e6=−e6,e5,e7=2e7,e5,e8=e8,e6,e7=e8,e7,e5,e8,e1,e2,e3,e6,e4
DGsetupLD1
Lie algebra: sl3a
We see that the grading weights are increasing in this new basis.
Tools:-DGinfotable,Grading
table−1=e2,e3,0=e4,e5,−2=e1,1=e6,e7,2=e8
With the keyword argument order = decreasing we obtain:
LD2,B2≔LieAlgebraDataG,sl3b,order=decreasing,output=basis
LD2,B2≔e1,e4=−2e1,e1,e5=−e1,e1,e6=−e3,e1,e7=e2,e1,e8=e4,e2,e3=e1,e2,e4=−e2,e2,e5=e2,e2,e6=e4−e5,e2,e8=−e7,e3,e4=−e3,e3,e5=−2e3,e3,e7=e5,e3,e8=e6,e4,e6=−e6,e4,e7=−e7,e4,e8=−2e8,e5,e6=e6,e5,e7=−2e7,e5,e8=−e8,e6,e7=−e8,e4,e3,e6,e1,e2,e5,e8,e7
DGsetupLD2
Lie algebra: sl3b
Now the grading weights are decreasing in the new basis.
table−1=e6,e7,0=e4,e5,−2=e8,1=e2,e3,2=e1
With the second calling sequence we can, for example, also obtain the structure equations for just the negatively graded components of sl3.
LD3,LD4,B3,B4≔LieAlgebraDataG,sl3c,negative,sl3minus,output=basis
LD3,LD4,B3,B4≔e1,e4=2e1,e1,e5=e1,e1,e6=e3,e1,e7=−e2,e1,e8=−e4,e2,e3=−e1,e2,e4=e2,e2,e5=−e2,e2,e6=e5−e4,e2,e8=e7,e3,e4=e3,e3,e5=2e3,e3,e7=−e5,e3,e8=−e6,e4,e6=e6,e4,e7=e7,e4,e8=2e8,e5,e6=−e6,e5,e7=2e7,e5,e8=e8,e6,e7=e8,e2,e3=−e1,e7,e5,e8,e1,e2,e3,e6,e4,e7,e5,e8
We see that the negatively graded components give the subalgebra:
B4
e7,e5,e8
with structure equations:
LD4
e2,e3=−e1
See Also
DifferentialGeometry
LieAlgebras
DGinfo
GradeSemiSimpleLieAlgebra
LieAlgebraData
SimpleLieAlgebraData
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