LieAlgebras
LieAlgebraData
convert different realizations of a Lie algebra to a Lie algebra data structure
Calling Sequence
Parameters
Description
Examples
LieAlgebraData(LieAlgebraPresentation)
LieAlgebraPresentation
-
one of several different formats for defining a Lie algebra
In the LieAlgebras package, the command DGsetup is used to initialize a Lie algebra -- that is, to define the basis elements for the Lie algebra and its dual and to store the structure constants for the Lie algebra in memory. The first argument for DGsetup is a Lie algebra data structure which contains the structure constants in a standard format used by the LieAlgebras package.
The purpose of the function LieAlgebraData is to convert various different presentations of a Lie algebra, which are commonly used in differential geometry and Lie theory, into the standard Lie algebra data structure required by the DGsetup command.
The types of Lie algebra presentations which can currently be converted to a Lie algebra data structure are:
FormStructureEquations
Grading
LieAlgebraName
MatrixAlgebra
StructureConstants
Subalgebra
VectorFields
VectorStructureEquations
The command LieAlgebraData returns a Lie algebra data structure. The structure equations defined by this data structure are displayed.
The command LieAlgebraData is part of the DifferentialGeometry:-LieAlgebras package. It can be used in the form LieAlgebraData(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-LieAlgebraData(...).
withDifferentialGeometry:withLieAlgebras:
Example 1. LieAlgebraData(StructureConstants)
In this example we create a 3 dimensional Array, C, of structure constants and use this Array to create a Lie algebra data structure for a Lie algebra called Ex1.
C≔Array1..3,1..3,1..3,0:
C1,3,1≔1:C3,1,1≔−1:C2,3,2≔1:C3,2,2≔−1:
L1≔LieAlgebraDataC,Ex1
L1≔e1,e3=e1,e2,e3=e2
Example 2. LieAlgebraData(VectorStructureEquations)
In this example we create a Lie algebra data structure for a Lie algebra called Ex2 from a list of structure equations for the Lie brackets.
VectStrEq≔x1,x3=x1,x2,x3=x1+x2,x1,x2,x3
L2≔LieAlgebraDataVectStrEq,Ex2
L2≔e1,e3=e1,e2,e3=e1+e2
Example 3. LieAlgebraData(FormStructureEquations)
In this example we create a Lie algebra data structure for a Lie algebra called Ex3 from a list of structure equations for the exterior derivatives of the dual 1-forms.
FormStrEq≔dθ1=−θ1&wθ2,dθ2=−θ2&wθ3,dθ3=0,θ1,θ2,θ3
L3≔LieAlgebraDataFormStrEq,Ex3
L3≔e1,e2=e1,e2,e3=e2
Example 4. LieAlgebraData(MatrixAlgebra)
In this example we create a Lie algebra data structure for a Lie algebra called Ex4 from a list of matrices.
MatrixAlg≔Matrix0,1,0,0,Matrix1,0,0,−1,Matrix0,0,1,0
MatrixAlg≔0100,100−1,0010
L4≔LieAlgebraDataMatrixAlg,Ex4
L4≔e1,e2=−2e1,e1,e3=e2,e2,e3=−2e3
Example 5. LieAlgebraData(LieAlgebraName)
In this example we create a Lie algebra data structure for a Lie algebra called Ex5 from a previously initialized Lie algebra.
First we initialize a Lie algebra Ex5.
L5≔_DGLieAlgebra,Ex5,3,1,2,1,1:
DGsetupL5
Lie algebra: Ex5
Applying LieAlgebraData to Ex5 gives back the Lie algebra data structure we started from.
LieAlgebraDataEx5,copyEx5
e1,e2=e1
Example 6. LieAlgebraData(Subalgebra, "LieAlgebraData")
In this example we create a Lie algebra data structure from a subalgebra of a Lie algebra.
First we initialize a Lie algebra Alg5.
L≔_DGLieAlgebra,Alg5,4,1,4,1,2,2,3,1,1,2,4,2,1,3,4,3,1;DGsetupL:
L≔e1,e4=2e1,e2,e3=e1,e2,e4=e2,e3,e4=e3
The vectors [e1, e2, e3] define a Lie subalgebra which we wish to initialize as a Lie algebra in its own right.
S1≔e1,e2,e3:
L6≔LieAlgebraDataS1,Ex6
L6≔e2,e3=e1
See Also
DifferentialGeometry
Query
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