Jacobi - Maple Help

Query[Jacobi] - check if a list of structure equations defines a Lie algebra by verifying the Jacobi identities

Calling Sequences

Query(Alg, "Jacobi")

Query(Alg, parm, "Jacobi")

Parameters

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

Description

 • A bracket operation $\left[\cdot ,\cdot \right]$ on a vector space defines a Lie bracket if it is bi-linear, skew-symmetric, and satisfies the Jacobi identity
 • In terms of the standard exterior derivative operator defined on the exterior algebra of the dual space (defined on 1-forms  by $(\mathrm{dω}$, the Jacobi identities are equivalent to the fundamental identity
 • 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(...).

Examples

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

Example 1.

We begin by defining a bracket operation on a 3-dimensional vector space with basis This bracket depends upon two parameters ${a}_{1}$ and ${a}_{2}$. We shall determine for which parameter values this bracket satisfies the Jacobi identities.

 > $\mathrm{Eq}≔\left[\left[\mathrm{x1},\mathrm{x2}\right]=\mathrm{a2}\mathrm{x2},\left[\mathrm{x1},\mathrm{x3}\right]=\mathrm{a1}\mathrm{x1}\right]:$

Convert to a Lie algebra data structure.

 > $L≔\mathrm{LieAlgebraData}\left(\mathrm{Eq},\left[\mathrm{x1},\mathrm{x2},\mathrm{x3}\right],\mathrm{Alg1}\right)$
 ${L}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{a2}}{}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{a1}}{}{\mathrm{e1}}\right]$ (2.1)

Initialize this data structure.

 > $\mathrm{DGsetup}\left(L\right)$
 ${\mathrm{Lie algebra: Alg1}}$ (2.2)
 Alg1 > $\mathrm{TF},\mathrm{EQ},\mathrm{SOLN},\mathrm{AlgList}≔\mathrm{Query}\left(\left\{\mathrm{a1},\mathrm{a2}\right\},"Jacobi"\right)$
 ${\mathrm{TF}}{,}{\mathrm{EQ}}{,}{\mathrm{SOLN}}{,}{\mathrm{AlgList}}{:=}{\mathrm{true}}{,}\left\{{0}{,}{-}{\mathrm{a1}}{}{\mathrm{a2}}\right\}{,}\left[\left\{{\mathrm{a1}}{=}{0}{,}{\mathrm{a2}}{=}{\mathrm{a2}}\right\}{,}\left\{{\mathrm{a1}}{=}{\mathrm{a1}}{,}{\mathrm{a2}}{=}{0}\right\}\right]{,}\left[\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{a2}}{}{\mathrm{e2}}\right]{,}\left[\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{a1}}{}{\mathrm{e1}}\right]\right]$ (2.3)

The equations that must be satisfied for the bracket to satisfy Jacobi are:

 Alg1 > $\mathrm{EQ}$
 $\left\{{0}{,}{-}{\mathrm{a1}}{}{\mathrm{a2}}\right\}$ (2.4)

This leads to two cases  or .  We initialize the resulting Lie algebra data structures and print the multiplication tables.

 Alg1 > $\mathrm{DGsetup}\left(\mathrm{AlgList}\left[1\right],\left[x\right],\left[\mathrm{\alpha }\right]\right):$$\mathrm{DGsetup}\left(\mathrm{AlgList}\left[2\right],\left[y\right],\left[\mathrm{\beta }\right]\right):$
 Alg1_2 > $\mathrm{print}\left(\mathrm{MultiplicationTable}\left("Alg1_1","LieBracket"\right),\mathrm{MultiplicationTable}\left("Alg1_2","LieBracket"\right)\right)$
 $\left[\left[{\mathrm{x1}}{,}{\mathrm{x2}}\right]{=}{\mathrm{a2}}{}{\mathrm{x2}}\right]{,}\left[\left[{\mathrm{y1}}{,}{\mathrm{y3}}\right]{=}{\mathrm{a1}}{}{\mathrm{y1}}\right]$ (2.5)

Example 2

The Jacobi identities are equivalent to the vanishing of the square of the exterior derivative.  For example:

 Alg1_2 > $\mathrm{ChangeLieAlgebraTo}\left(\mathrm{Alg1}\right):$
 Alg1 > $\mathrm{ExteriorDerivative}\left(\mathrm{ExteriorDerivative}\left(\mathrm{θ1}\right)\right)$
 ${0}{}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ3}}$ (2.6)
 Alg1 > $\mathrm{ExteriorDerivative}\left(\mathrm{ExteriorDerivative}\left(\mathrm{θ2}\right)\right)$
 ${-}{\mathrm{a1}}{}{\mathrm{a2}}{}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ3}}$ (2.7)
 Alg1 > $\mathrm{ExteriorDerivative}\left(\mathrm{ExteriorDerivative}\left(\mathrm{θ3}\right)\right)$
 ${0}{}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ3}}$ (2.8)