DifferentialGeometry/LieAlgebras/LieAlgebraData/Grading - Maple Help
For the best experience, we recommend viewing online help using Google Chrome or Microsoft Edge.

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : DifferentialGeometry/LieAlgebras/LieAlgebraData/Grading

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

Description

• 

 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 = 𝔤1 𝔤0  𝔤1 with dim 𝔤1 = 2, dim 𝔤0 = 3, dim 𝔤1 = 1 then e1, e2, e3, e4, e5, e6 is an adapted basis, in increasing ordering of weights, if 𝔤1= 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 𝔤1 = 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.

Examples

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

LDe1,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=e1e2,e3,e6=e4,e3,e7=e8,e4,e5=e6,e4,e7=e1,e4,e8=e3,e5,e8=e7,e6,e7=e5,e6,e8=e2

(2.1)

DGsetupLD

Lie algebra: sl3

(2.2)

 

Here is the grading we shall use.

sl3 > 

G ≔ table0=e1,e2,1=e3,e6,2=e4,2=e7,1=e5,e8

Gtable−1=_DGvector,sl3,,5,1,_DGvector,sl3,,5,1,_DGvector,sl3,,5,1,_DGvector,sl3,,5,1,_DGvector,sl3,,5,1,_DGvector,sl3,,5,1,_DGvector,sl3,,5,1,_DGvector,sl3,,5,1,_DGvector,sl3,,5,1,_DGvector,sl3,,5,1,_DGvector,sl3,,5,1,_DGvector,sl3,,5,1,_DGvector,sl3,,5,1,_DGvector,sl3,,5,1,_DGvector,sl3,,5,1,_DGvector,sl3,,5,1,_DGvector,sl3,,8,1,_DGvector,sl3,,8,1,_DGvector,sl3,,8,1,_DGvector,sl3,,8,1,_DGvector,sl3,,8,1,_DGvector,sl3,,8,1,_DGvector,sl3,,8,1,_DGvector,sl3,,8,1,_DGvector,sl3,,8,1,_DGvector,sl3,,8,1,_DGvector,sl3,,8,1,_DGvector,sl3,,8,1,_DGvector,sl3,,8,1,_DGvector,sl3,,8,1,_DGvector,sl3,,8,1,_DGvector,sl3,,8,1,0=_DGvector,sl3,,1,1,_DGvector,sl3,,1,1,_DGvector,sl3,,1,1,_DGvector,sl3,,1,1,_DGvector,sl3,,1,1,_DGvector,sl3,,1,1,_DGvector,sl3,,1,1,_DGvector,sl3,,1,1,_DGvector,sl3,,1,1,_DGvector,sl3,,1,1,_DGvector,sl3,,1,1,_DGvector,sl3,,1,1,_DGvector,sl3,,1,1,_DGvector,sl3,,1,1,_DGvector,sl3,,1,1,_DGvector,sl3,,1,1,_DGvector,sl3,,2,1,_DGvector,sl3,,2,1,_DGvector,sl3,,2,1,_DGvector,sl3,,2,1,_DGvector,sl3,,2,1,_DGvector,sl3,,2,1,_DGvector,sl3,,2,1,_DGvector,sl3,,2,1,_DGvector,sl3,,2,1,_DGvector,sl3,,2,1,_DGvector,sl3,,2,1,_DGvector,sl3,,2,1,_DGvector,sl3,,2,1,_DGvector,sl3,,2,1,_DGvector,sl3,,2,1,_DGvector,sl3,,2,1,−2=_DGvector,sl3,,7,1,_DGvector,sl3,,7,1,_DGvector,sl3,,7,1,_DGvector,sl3,,7,1,_DGvector,sl3,,7,1,_DGvector,sl3,,7,1,_DGvector,sl3,,7,1,_DGvector,sl3,,7,1,_DGvector,sl3,,7,1,_DGvector,sl3,,7,1,_DGvector,sl3,,7,1,_DGvector,sl3,,7,1,_DGvector,sl3,,7,1,_DGvector,sl3,,7,1,_DGvector,sl3,,7,1,_DGvector,sl3,,7,1,1=_DGvector,sl3,,3,1,_DGvector,sl3,,3,1,_DGvector,sl3,,3,1,_DGvector,sl3,,3,1,_DGvector,sl3,,3,1,_DGvector,sl3,,3,1,_DGvector,sl3,,3,1,_DGvector,sl3,,3,1,_DGvector,sl3,,3,1,_DGvector,sl3,,3,1,_DGvector,sl3,,3,1,_DGvector,sl3,,3,1,_DGvector,sl3,,3,1,_DGvector,sl3,,3,1,_DGvector,sl3,,3,1,_DGvector,sl3,,3,1,_DGvector,sl3,,6,1,_DGvector,sl3,,6,1,_DGvector,sl3,,6,1,_DGvector,sl3,,6,1,_DGvector,sl3,,6,1,_DGvector,sl3,,6,1,_DGvector,sl3,,6,1,_DGvector,sl3,,6,1,_DGvector,sl3,,6,1,_DGvector,sl3,,6,1,_DGvector,sl3,,6,1,_DGvector,sl3,,6,1,_DGvector,sl3,,6,1,_DGvector,sl3,,6,1,_DGvector,sl3,,6,1,_DGvector,sl3,,6,1,2=_DGvector,sl3,,4,1,_DGvector,sl3,,4,1,_DGvector,sl3,,4,1,_DGvector,sl3,,4,1,_DGvector,sl3,,4,1,_DGvector,sl3,,4,1,_DGvector,sl3,,4,1,_DGvector,sl3,,4,1,_DGvector,sl3,,4,1,_DGvector,sl3,,4,1,_DGvector,sl3,,4,1,_DGvector,sl3,,4,1,_DGvector,sl3,,4,1,_DGvector,sl3,,4,1,_DGvector,sl3,,4,1,_DGvector,sl3,,4,1

(2.3)

 

Here are the structure equations for sl3 adapted to the basis e7, e5,e8,e1, e2, e3, e6, e4.

sl3 > 

LD1,B1 ≔ LieAlgebraDataG,sl3a,output=basis

LD1,B1e1,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=e5e4,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,_DGvector,sl3,,7,1,_DGvector,sl3,,7,1,_DGvector,sl3,,7,1,_DGvector,sl3,,7,1,_DGvector,sl3,,7,1,_DGvector,sl3,,7,1,_DGvector,sl3,,7,1,_DGvector,sl3,,7,1,_DGvector,sl3,,7,1,_DGvector,sl3,,7,1,_DGvector,sl3,,7,1,_DGvector,sl3,,7,1,_DGvector,sl3,,7,1,_DGvector,sl3,,7,1,_DGvector,sl3,,7,1,_DGvector,sl3,,7,1,_DGvector,sl3,,5,1,_DGvector,sl3,,5,1,_DGvector,sl3,,5,1,_DGvector,sl3,,5,1,_DGvector,sl3,,5,1,_DGvector,sl3,,5,1,_DGvector,sl3,,5,1,_DGvector,sl3,,5,1,_DGvector,sl3,,5,1,_DGvector,sl3,,5,1,_DGvector,sl3,,5,1,_DGvector,sl3,,5,1,_DGvector,sl3,,5,1,_DGvector,sl3,,5,1,_DGvector,sl3,,5,1,_DGvector,sl3,,5,1,_DGvector,sl3,,8,1,_DGvector,sl3,,8,1,_DGvector,sl3,,8,1,_DGvector,sl3,,8,1,_DGvector,sl3,,8,1,_DGvector,sl3,,8,1,_DGvector,sl3,,8,1,_DGvector,sl3,,8,1,_DGvector,sl3,,8,1,_DGvector,sl3,,8,1,_DGvector,sl3,,8,1,_DGvector,sl3,,8,1,_DGvector,sl3,,8,1,_DGvector,sl3,,8,1,_DGvector,sl3,,8,1,_DGvector,sl3,,8,1,_DGvector,sl3,,1,1,_DGvector,sl3,,1,1,_DGvector,sl3,,1,1,_DGvector,sl3,,1,1,_DGvector,sl3,,1,1,_DGvector,sl3,,1,1,_DGvector,sl3,,1,1,_DGvector,sl3,,1,1,_DGvector,sl3,,1,1,_DGvector,sl3,,1,1,_DGvector,sl3,,1,1,_DGvector,sl3,,1,1,_DGvector,sl3,,1,1,_DGvector,sl3,,1,1,_DGvector,sl3,,1,1,_DGvector,sl3,,1,1,_DGvector,sl3,,2,1,_DGvector,sl3,,2,1,_DGvector,sl3,,2,1,_DGvector,sl3,,2,1,_DGvector,sl3,,2,1,_DGvector,sl3,,2,1,_DGvector,sl3,,2,1,_DGvector,sl3,,2,1,_DGvector,sl3,,2,1,_DGvector,sl3,,2,1,_DGvector,sl3,,2,1,_DGvector,sl3,,2,1,_DGvector,sl3,,2,1,_DGvector,sl3,,2,1,_DGvector,sl3,,2,1,_DGvector,sl3,,2,1,_DGvector,sl3,,3,1,_DGvector,sl3,,3,1,_DGvector,sl3,,3,1,_DGvector,sl3,,3,1,_DGvector,sl3,,3,1,_DGvector,sl3,,3,1,_DGvector,sl3,,3,1,_DGvector,sl3,,3,1,_DGvector,sl3,,3,1,_DGvector,sl3,,3,1,_DGvector,sl3,,3,1,_DGvector,sl3,,3,1,_DGvector,sl3,,3,1,_DGvector,sl3,,3,1,_DGvector,sl3,,3,1,_DGvector,sl3,,3,1,_DGvector,sl3,,6,1,_DGvector,sl3,,6,1,_DGvector,sl3,,6,1,_DGvector,sl3,,6,1,_DGvector,sl3,,6,1,_DGvector,sl3,,6,1,_DGvector,sl3,,6,1,_DGvector,sl3,,6,1,_DGvector,sl3,,6,1,_DGvector,sl3,,6,1,_DGvector,sl3,,6,1,_DGvector,sl3,,6,1,_DGvector,sl3,,6,1,_DGvector,sl3,,6,1,_DGvector,sl3,,6,1,_DGvector,sl3,,6,1,_DGvector,sl3,,4,1,_DGvector,sl3,,4,1,_DGvector,sl3,,4,1,_DGvector,sl3,,4,1,_DGvector,sl3,,4,1,_DGvector,sl3,,4,1,_DGvector,sl3,,4,1,_DGvector,sl3,,4,1,_DGvector,sl3,,4,1,_DGvector,sl3,,4,1,_DGvector,sl3,,4,1,_DGvector,sl3,,4,1,_DGvector,sl3,,4,1,_DGvector,sl3,,4,1,_DGvector,sl3,,4,1,_DGvector,sl3,,4,1

(2.4)
sl3 > 

DGsetupLD1

Lie algebra: sl3a

(2.5)

 

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

sl3a > 

Tools:-DGinfotable,Grading

table−1=_DGvector,sl3a,,2,1,_DGvector,sl3a,,2,1,_DGvector,sl3a,,2,1,_DGvector,sl3a,,2,1,_DGvector,sl3a,,2,1,_DGvector,sl3a,,2,1,_DGvector,sl3a,,2,1,_DGvector,sl3a,,2,1,_DGvector,sl3a,,2,1,_DGvector,sl3a,,2,1,_DGvector,sl3a,,2,1,_DGvector,sl3a,,2,1,_DGvector,sl3a,,2,1,_DGvector,sl3a,,2,1,_DGvector,sl3a,,2,1,_DGvector,sl3a,,2,1,_DGvector,sl3a,,3,1,_DGvector,sl3a,,3,1,_DGvector,sl3a,,3,1,_DGvector,sl3a,,3,1,_DGvector,sl3a,,3,1,_DGvector,sl3a,,3,1,_DGvector,sl3a,,3,1,_DGvector,sl3a,,3,1,_DGvector,sl3a,,3,1,_DGvector,sl3a,,3,1,_DGvector,sl3a,,3,1,_DGvector,sl3a,,3,1,_DGvector,sl3a,,3,1,_DGvector,sl3a,,3,1,_DGvector,sl3a,,3,1,_DGvector,sl3a,,3,1,0=_DGvector,sl3a,,4,1,_DGvector,sl3a,,4,1,_DGvector,sl3a,,4,1,_DGvector,sl3a,,4,1,_DGvector,sl3a,,4,1,_DGvector,sl3a,,4,1,_DGvector,sl3a,,4,1,_DGvector,sl3a,,4,1,_DGvector,sl3a,,4,1,_DGvector,sl3a,,4,1,_DGvector,sl3a,,4,1,_DGvector,sl3a,,4,1,_DGvector,sl3a,,4,1,_DGvector,sl3a,,4,1,_DGvector,sl3a,,4,1,_DGvector,sl3a,,4,1,_DGvector,sl3a,,5,1,_DGvector,sl3a,,5,1,_DGvector,sl3a,,5,1,_DGvector,sl3a,,5,1,_DGvector,sl3a,,5,1,_DGvector,sl3a,,5,1,_DGvector,sl3a,,5,1,_DGvector,sl3a,,5,1,_DGvector,sl3a,,5,1,_DGvector,sl3a,,5,1,_DGvector,sl3a,,5,1,_DGvector,sl3a,,5,1,_DGvector,sl3a,,5,1,_DGvector,sl3a,,5,1,_DGvector,sl3a,,5,1,_DGvector,sl3a,,5,1,−2=_DGvector,sl3a,,1,1,_DGvector,sl3a,,1,1,_DGvector,sl3a,,1,1,_DGvector,sl3a,,1,1,_DGvector,sl3a,,1,1,_DGvector,sl3a,,1,1,_DGvector,sl3a,,1,1,_DGvector,sl3a,,1,1,_DGvector,sl3a,,1,1,_DGvector,sl3a,,1,1,_DGvector,sl3a,,1,1,_DGvector,sl3a,,1,1,_DGvector,sl3a,,1,1,_DGvector,sl3a,,1,1,_DGvector,sl3a,,1,1,_DGvector,sl3a,,1,1,1=_DGvector,sl3a,,6,1,_DGvector,sl3a,,6,1,_DGvector,sl3a,,6,1,_DGvector,sl3a,,6,1,_DGvector,sl3a,,6,1,_DGvector,sl3a,,6,1,_DGvector,sl3a,,6,1,_DGvector,sl3a,,6,1,_DGvector,sl3a,,6,1,_DGvector,sl3a,,6,1,_DGvector,sl3a,,6,1,_DGvector,sl3a,,6,1,_DGvector,sl3a,,6,1,_DGvector,sl3a,,6,1,_DGvector,sl3a,,6,1,_DGvector,sl3a,,6,1,_DGvector,sl3a,,7,1,_DGvector,sl3a,,7,1,_DGvector,sl3a,,7,1,_DGvector,sl3a,,7,1,_DGvector,sl3a,,7,1,_DGvector,sl3a,,7,1,_DGvector,sl3a,,7,1,_DGvector,sl3a,,7,1,_DGvector,sl3a,,7,1,_DGvector,sl3a,,7,1,_DGvector,sl3a,,7,1,_DGvector,sl3a,,7,1,_DGvector,sl3a,,7,1,_DGvector,sl3a,,7,1,_DGvector,sl3a,,7,1,_DGvector,sl3a,,7,1,2=_DGvector,sl3a,,8,1,_DGvector,sl3a,,8,1,_DGvector,sl3a,,8,1,_DGvector,sl3a,,8,1,_DGvector,sl3a,,8,1,_DGvector,sl3a,,8,1,_DGvector,sl3a,,8,1,_DGvector,sl3a,,8,1,_DGvector,sl3a,,8,1,_DGvector,sl3a,,8,1,_DGvector,sl3a,,8,1,_DGvector,sl3a,,8,1,_DGvector,sl3a,,8,1,_DGvector,sl3a,,8,1,_DGvector,sl3a,,8,1,_DGvector,sl3a,,8,1

(2.6)

 

With the keyword argument order = decreasing we obtain:

sl3a > 

LD2,B2 ≔ LieAlgebraDataG,sl3b,order=decreasing,output=basis

LD2,B2e1&comma;e4=2e1&comma;e1&comma;e5=e1&comma;e1&comma;e6=e3&comma;e1&comma;e7=e2&comma;e1&comma;e8=e4&comma;e2&comma;e3=e1&comma;e2&comma;e4=e2&comma;e2&comma;e5=e2&comma;e2&comma;e6=e4e5&comma;e2&comma;e8=e7&comma;e3&comma;e4=e3&comma;e3&comma;e5=2e3&comma;e3&comma;e7=e5&comma;e3&comma;e8=e6&comma;e4&comma;e6=e6&comma;e4&comma;e7=e7&comma;e4&comma;e8=2e8&comma;e5&comma;e6=e6&comma;e5&comma;e7=2e7&comma;e5&comma;e8=e8&comma;e6&comma;e7=e8,_DGvector&comma;sl3&comma;&comma;4&comma;1&comma;_DGvector&comma;sl3&comma;&comma;4&comma;1&comma;_DGvector&comma;sl3&comma;&comma;4&comma;1&comma;_DGvector&comma;sl3&comma;&comma;4&comma;1&comma;_DGvector&comma;sl3&comma;&comma;4&comma;1&comma;_DGvector&comma;sl3&comma;&comma;4&comma;1&comma;_DGvector&comma;sl3&comma;&comma;4&comma;1&comma;_DGvector&comma;sl3&comma;&comma;4&comma;1&comma;_DGvector&comma;sl3&comma;&comma;4&comma;1&comma;_DGvector&comma;sl3&comma;&comma;4&comma;1&comma;_DGvector&comma;sl3&comma;&comma;4&comma;1&comma;_DGvector&comma;sl3&comma;&comma;4&comma;1&comma;_DGvector&comma;sl3&comma;&comma;4&comma;1&comma;_DGvector&comma;sl3&comma;&comma;4&comma;1&comma;_DGvector&comma;sl3&comma;&comma;4&comma;1&comma;_DGvector&comma;sl3&comma;&comma;4&comma;1&comma;_DGvector&comma;sl3&comma;&comma;3&comma;1&comma;_DGvector&comma;sl3&comma;&comma;3&comma;1&comma;_DGvector&comma;sl3&comma;&comma;3&comma;1&comma;_DGvector&comma;sl3&comma;&comma;3&comma;1&comma;_DGvector&comma;sl3&comma;&comma;3&comma;1&comma;_DGvector&comma;sl3&comma;&comma;3&comma;1&comma;_DGvector&comma;sl3&comma;&comma;3&comma;1&comma;_DGvector&comma;sl3&comma;&comma;3&comma;1&comma;_DGvector&comma;sl3&comma;&comma;3&comma;1&comma;_DGvector&comma;sl3&comma;&comma;3&comma;1&comma;_DGvector&comma;sl3&comma;&comma;3&comma;1&comma;_DGvector&comma;sl3&comma;&comma;3&comma;1&comma;_DGvector&comma;sl3&comma;&comma;3&comma;1&comma;_DGvector&comma;sl3&comma;&comma;3&comma;1&comma;_DGvector&comma;sl3&comma;&comma;3&comma;1&comma;_DGvector&comma;sl3&comma;&comma;3&comma;1&comma;_DGvector&comma;sl3&comma;&comma;6&comma;1&comma;_DGvector&comma;sl3&comma;&comma;6&comma;1&comma;_DGvector&comma;sl3&comma;&comma;6&comma;1&comma;_DGvector&comma;sl3&comma;&comma;6&comma;1&comma;_DGvector&comma;sl3&comma;&comma;6&comma;1&comma;_DGvector&comma;sl3&comma;&comma;6&comma;1&comma;_DGvector&comma;sl3&comma;&comma;6&comma;1&comma;_DGvector&comma;sl3&comma;&comma;6&comma;1&comma;_DGvector&comma;sl3&comma;&comma;6&comma;1&comma;_DGvector&comma;sl3&comma;&comma;6&comma;1&comma;_DGvector&comma;sl3&comma;&comma;6&comma;1&comma;_DGvector&comma;sl3&comma;&comma;6&comma;1&comma;_DGvector&comma;sl3&comma;&comma;6&comma;1&comma;_DGvector&comma;sl3&comma;&comma;6&comma;1&comma;_DGvector&comma;sl3&comma;&comma;6&comma;1&comma;_DGvector&comma;sl3&comma;&comma;6&comma;1&comma;_DGvector&comma;sl3&comma;&comma;1&comma;1&comma;_DGvector&comma;sl3&comma;&comma;1&comma;1&comma;_DGvector&comma;sl3&comma;&comma;1&comma;1&comma;_DGvector&comma;sl3&comma;&comma;1&comma;1&comma;_DGvector&comma;sl3&comma;&comma;1&comma;1&comma;_DGvector&comma;sl3&comma;&comma;1&comma;1&comma;_DGvector&comma;sl3&comma;&comma;1&comma;1&comma;_DGvector&comma;sl3&comma;&comma;1&comma;1&comma;_DGvector&comma;sl3&comma;&comma;1&comma;1&comma;_DGvector&comma;sl3&comma;&comma;1&comma;1&comma;_DGvector&comma;sl3&comma;&comma;1&comma;1&comma;_DGvector&comma;sl3&comma;&comma;1&comma;1&comma;_DGvector&comma;sl3&comma;&comma;1&comma;1&comma;_DGvector&comma;sl3&comma;&comma;1&comma;1&comma;_DGvector&comma;sl3&comma;&comma;1&comma;1&comma;_DGvector&comma;sl3&comma;&comma;1&comma;1&comma;_DGvector&comma;sl3&comma;&comma;2&comma;1&comma;_DGvector&comma;sl3&comma;&comma;2&comma;1&comma;_DGvector&comma;sl3&comma;&comma;2&comma;1&comma;_DGvector&comma;sl3&comma;&comma;2&comma;1&comma;_DGvector&comma;sl3&comma;&comma;2&comma;1&comma;_DGvector&comma;sl3&comma;&comma;2&comma;1&comma;_DGvector&comma;sl3&comma;&comma;2&comma;1&comma;_DGvector&comma;sl3&comma;&comma;2&comma;1&comma;_DGvector&comma;sl3&comma;&comma;2&comma;1&comma;_DGvector&comma;sl3&comma;&comma;2&comma;1&comma;_DGvector&comma;sl3&comma;&comma;2&comma;1&comma;_DGvector&comma;sl3&comma;&comma;2&comma;1&comma;_DGvector&comma;sl3&comma;&comma;2&comma;1&comma;_DGvector&comma;sl3&comma;&comma;2&comma;1&comma;_DGvector&comma;sl3&comma;&comma;2&comma;1&comma;_DGvector&comma;sl3&comma;&comma;2&comma;1&comma;_DGvector&comma;sl3&comma;&comma;5&comma;1&comma;_DGvector&comma;sl3&comma;&comma;5&comma;1&comma;_DGvector&comma;sl3&comma;&comma;5&comma;1&comma;_DGvector&comma;sl3&comma;&comma;5&comma;1&comma;_DGvector&comma;sl3&comma;&comma;5&comma;1&comma;_DG<