Query[Jacobi] - check if a list of structure equations defines a Lie algebra by verifying the Jacobi identities
Query(Alg, parm, "Jacobi")
Alg - (optional) the name of an initialized Lie algebra
parm - (optional) a set of parameters appearing in the structure equations of the Lie algebra g
A bracket operation ⋅,⋅ on a vector space 𝔤 defines a Lie bracket if it is bi-linear, skew-symmetric, and satisfies the Jacobi identity x,y, z +z,x,y + y,z,x =0.
In terms of the standard exterior derivative operator d defined on the exterior algebra of the dual space 𝔤 *(defined on 1-forms ω ∈ 𝔤* by (dω)x,y = − ωx,y, the Jacobi identities are equivalent to the fundamental identity d2 =0 .
The program DGsetup does not check that its input, a Lie algebra data structure, actually defines a Lie algebra. To verify that a Lie algebra data structure does indeed define a Lie algebra, initialize the Lie algebra data structure, and run Query("Jacobi").
Query(Alg, "Jacobi") returns true if the Jacobi identities hold (in which case Alg defines a Lie algebra) and false otherwise. If the algebra is unspecified, then Query is applied to the current algebra. The Jacobi identity is checked using the exterior derivative formulation.
Query(Alg, parm, "Jacobi") returns a sequence TF, Eq, Soln, AlgList. Here TF is true if Maple finds parameter values for which the Jacobi identities are valid and false otherwise; Eq is the set of equations (with the variables parm as unknowns) which must be satisfied for the Jacobi identities to hold; Soln is the list of solutions to the equations Eq; and AlgList is the list of Lie algebra data structures obtained from the parameter values given by various solutions in Soln.
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(...).
We begin by defining a bracket operation on a 3-dimensional vector space with basis x1, x2, x3. This bracket depends upon two parameters a1 and a2. We shall determine for which parameter values this bracket satisfies the Jacobi identities.
Eq ≔ x1,x2=a2⁢x2,x1,x3=a1⁢x1:
Convert to a Lie algebra data structure.
L ≔ LieAlgebraData⁡Eq,x1,x2,x3,Alg1
Initialize this data structure.
Lie algebra: Alg1
TF,EQ,SOLN,AlgList ≔ Query⁡a1,a2,Jacobi
The equations that must be satisfied for the bracket to satisfy Jacobi are:
This leads to two cases a1 =0 or a2 =0. We initialize the resulting Lie algebra data structures and print the multiplication tables.
The Jacobi identities are equivalent to the vanishing of the square of the exterior derivative. For example:
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