 Deformation - Maple Help

LieAlgebras[Deformation] - find the deformation of a Lie algebra defined by a list of 2-forms

Calling Sequences

Deformation(Alg$,$${\mathrm{Ω}}$$,$tAlgName$,$option)

Parameters

Alg     - the name of an initialized Lie algebra $\mathrm{𝔤}$

a list of 2-forms on with values in $\mathrm{𝔤}$

t       - an unassigned name to be used as the deformation parameter, or a list of unassigned names

AlgName - an unassigned name (or string) for the deformation algebra

option  - the keyword argument parameters = [a, b, ... ] Description

 • Let be a finite-dimensional Lie algebra. A deformation of $\mathrm{𝔤}$ is a smoothly varying family of Lie algebras (all of the same dimension) such that $\mathrm{𝔤}$The deformation is called trivial if the Lie algebras are isomorphic for all values of $t$. Deformations are calculated as a formal power series for the bracket operation in

Here and each coefficient is a bilinear, skew-symmetric mapping : , that is, . The Jacobi identity for the bracket  imposes a set of conditions on the coefficients These conditions are described below in equations (1), (2) and (3).

 • The command Deformation will return the structure equations for the bracket operation using the Lie bracket $\left[x,y\right]$ defined by the first argument Alg and the forms  given by the second argument. The procedure Deformation does not verify that the forms satisfy the conditions (1), (2) and (3) below so that the bracket operation need not satisfy the Jacobi identity.
 • Suppose that the forms depend upon a number of parameters With the keyword argument parameters = [a, b, ...], the procedure Deformation initializes the deformation algebra defined by  (using the name provided by the 4th argument) and calculates the conditions on these parameters imposed by the Jacobi identities. A sequence TF, Eq,  Soln, LD of 4 elements is returned, where TF is true if there is a set of parameter values satisfying the Jacobi identities, Eq is the set of equations arising from the Jacobi equations, Soln is the list of solutions to the Jacobi equations for the parameters  and LD the Lie algebra data structures defined by these solutions.
 • The conditions imposed on the coefficients by the Jacobi identity for the bracket  are as follows.First, the 2-form must be closed, that is,

${\mathrm{dη}}_{1}$ = 0,      (1)

where is the exterior derivative operator. If is an exact form, that is, , then the linear deformation is a trivial deformation. Thus, to determine the possible non-trivial deformations, one first computes the cohomology . This can be done with the commands Representation, RelativeChains, Cohomology.

 • If the Massey product $\left[{\mathrm{η}}_{1},{\mathrm{η}}_{1}\right]$ of the linear deformation vanishes, then the Jacobi identity holds and the linear deformation  

defines a Lie algebra. Otherwise, the quadratic deformation ${\mathrm{η}}_{2}$ can be determined by the equation

$\left[{\mathrm{η}}_{1},{\mathrm{η}}_{1}\right]$ = 0.      (2)

This implies that for the quadratic deformation to exist, the Massey product $\left[{\mathrm{η}}_{1},{\mathrm{η}}_{1}\right]$ must be an exact 2-form. The quadratic deformation can be found using the command CohomologyDecomposition. The higher order deformations are determined by the equations

$\left[{\mathrm{η}}_{1},{\mathrm{η}}_{2}\right]$ = 0,    $\left[{\mathrm{η}}_{1},{\mathrm{η}}_{3}\right]$ = 0,  etc.  (3)

 • See D. B. Fuks Cohomology of Infinite Dimensional Lie Algebras (pages 35 - 38) for more details on deformations of Lie algebras and other applications of Lie algebra cohomology. Examples

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

First initialize an 8-dimensional Lie algebra. We shall create various deformations of this Lie algebra. Here are the structure equations.

 > $\mathrm{StrEq}≔\left[\left[\mathrm{x4},\mathrm{x6}\right]=-\mathrm{x3},\left[\mathrm{x4},\mathrm{x7}\right]=-\mathrm{x1},\left[\mathrm{x5},\mathrm{x6}\right]=-\mathrm{x1},\left[\mathrm{x5},\mathrm{x7}\right]=-\mathrm{x2},\left[\mathrm{x6},\mathrm{x8}\right]=-\mathrm{x4},\left[\mathrm{x7},\mathrm{x8}\right]=-\mathrm{x5}\right]$
 ${\mathrm{StrEq}}{:=}\left[\left[{\mathrm{x4}}{,}{\mathrm{x6}}\right]{=}{-}{\mathrm{x3}}{,}\left[{\mathrm{x4}}{,}{\mathrm{x7}}\right]{=}{-}{\mathrm{x1}}{,}\left[{\mathrm{x5}}{,}{\mathrm{x6}}\right]{=}{-}{\mathrm{x1}}{,}\left[{\mathrm{x5}}{,}{\mathrm{x7}}\right]{=}{-}{\mathrm{x2}}{,}\left[{\mathrm{x6}}{,}{\mathrm{x8}}\right]{=}{-}{\mathrm{x4}}{,}\left[{\mathrm{x7}}{,}{\mathrm{x8}}\right]{=}{-}{\mathrm{x5}}\right]$ (2.1)

Use the commands LieAlgebraData and DGsetup to initialize this Lie algebra.

 > $\mathrm{LD}≔\mathrm{LieAlgebraData}\left(\mathrm{StrEq},\left[\mathrm{x1},\mathrm{x2},\mathrm{x3},\mathrm{x4},\mathrm{x5},\mathrm{x6},\mathrm{x7},\mathrm{x8}\right],\mathrm{alg}\right)$
 ${\mathrm{LD}}{:=}\left[\left[{\mathrm{e4}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e5}}\right]$ (2.2)
 > $\mathrm{DGsetup}\left(\mathrm{LD}\right)$
 ${\mathrm{Lie algebra: alg}}$ (2.3)

We also need a vector space on which we can define the adjoint representation (See Adjoint and Representation).

 alg > $\mathrm{DGsetup}\left(\left[\mathrm{w1},\mathrm{w2},\mathrm{w3},\mathrm{w4},\mathrm{w5},\mathrm{w6},\mathrm{w7},\mathrm{w8}\right],W\right)$
 ${\mathrm{frame name: W}}$ (2.4)
 alg > $\mathrm{ρ}≔\mathrm{Representation}\left(\mathrm{alg},W,\mathrm{Adjoint}\left(\mathrm{alg}\right)\right):$
 alg > $\mathrm{DGsetup}\left(\mathrm{alg},\mathrm{ρ},\mathrm{algW}\right)$
 ${\mathrm{Lie algebra with coefficients: algW}}$ (2.5)

The linear deformations are given in terms of the Lie algebra cohomology of $\mathrm{𝔤}$ with coefficients in the adjoint representation. This cohomology is computed to be:

 > $\mathrm{H2}≔\mathrm{evalDG}\left(\left[\mathrm{w2}\mathrm{θ3}&w\mathrm{θ6},\mathrm{w2}\mathrm{θ1}&w\mathrm{θ6}-2\mathrm{w1}\mathrm{θ3}&w\mathrm{θ6}+\mathrm{w2}\mathrm{θ3}&w\mathrm{θ7},\mathrm{w3}\mathrm{θ1}&w\mathrm{θ6}-2\mathrm{w1}\mathrm{θ1}&w\mathrm{θ7}-2\mathrm{w1}\mathrm{θ2}&w\mathrm{θ6}+\mathrm{w2}\mathrm{θ2}&w\mathrm{θ7}+\mathrm{w3}\mathrm{θ3}&w\mathrm{θ7},\mathrm{w3}\mathrm{θ1}&w\mathrm{θ7}+\mathrm{w3}\mathrm{θ2}&w\mathrm{θ6}-2\mathrm{w1}\mathrm{θ2}&w\mathrm{θ7},\mathrm{w3}\mathrm{θ2}&w\mathrm{θ7},-2\mathrm{w1}\mathrm{θ1}&w\mathrm{θ6}+\mathrm{w2}\mathrm{θ1}&w\mathrm{θ7}+\mathrm{w2}\mathrm{θ2}&w\mathrm{θ6}+\mathrm{w3}\mathrm{θ3}&w\mathrm{θ6}-2\mathrm{w1}\mathrm{θ3}&w\mathrm{θ7},\mathrm{w8}\mathrm{θ6}&w\mathrm{θ7}\right]\right)$
 ${\mathrm{H2}}{:=}\left[{\mathrm{w2}}{}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ6}}{,}{\mathrm{w2}}{}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ6}}{-}{2}{}{\mathrm{w1}}{}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ6}}{+}{\mathrm{w2}}{}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ7}}{,}{\mathrm{w3}}{}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ6}}{-}{2}{}{\mathrm{w1}}{}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ7}}{-}{2}{}{\mathrm{w1}}{}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ6}}{+}{\mathrm{w2}}{}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ7}}{+}{\mathrm{w3}}{}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ7}}{,}{\mathrm{w3}}{}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ7}}{+}{\mathrm{w3}}{}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ6}}{-}{2}{}{\mathrm{w1}}{}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ7}}{,}{\mathrm{w3}}{}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ7}}{,}{-}{2}{}{\mathrm{w1}}{}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ6}}{+}{\mathrm{w2}}{}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ7}}{+}{\mathrm{w2}}{}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ6}}{+}{\mathrm{w3}}{}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ6}}{-}{2}{}{\mathrm{w1}}{}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ7}}{,}{\mathrm{w8}}{}{\mathrm{θ6}}{}{\bigwedge }{}{\mathrm{θ7}}\right]$ (2.6)

We note that the 2-forms in ${H}^{2}$ are all closed.

 > $\mathrm{ExteriorDerivative}\left(\mathrm{H2}\right)$
 $\left[{0}{}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ3}}{,}{0}{}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ3}}{,}{0}{}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ3}}{,}{0}{}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ3}}{,}{0}{}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ3}}{,}{0}{}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ3}}{,}{0}{}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ3}}\right]$ (2.7)

Example 1.

We consider the Lie algebra deformation defined by the first cohomology class, represented by $\mathrm{H2}\left[1\right]$.

 algD1 > $\mathrm{η1}≔{\mathrm{H2}}_{1}$
 ${\mathrm{η1}}{:=}{\mathrm{w2}}{}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ6}}$ (2.8)
 algW > $\mathrm{LD1}≔\mathrm{Deformation}\left(\mathrm{alg},\left[\mathrm{η1}\right],\mathrm{κ},\mathrm{algD1}\right)$
 ${\mathrm{LD1}}{:=}\left[\left[{\mathrm{e3}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{κ}}{}{\mathrm{e2}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e5}}\right]$ (2.9)
 alg > $\mathrm{DGsetup}\left(\mathrm{LD1}\right)$
 ${\mathrm{Lie algebra: algD1}}$ (2.10)

We use the Query command to check that this deformation defines a Lie algebra.

 algD1 > $\mathrm{Query}\left("Jacobi"\right)$
 ${\mathrm{true}}$ (2.11)

Example 2.

Here we look at the Lie algebra deformation defined by the third cohomology class, represented by $\mathrm{H2}\left[3\right].$

 algD1 > $\mathrm{η1}≔{\mathrm{H2}}_{3}$
 ${\mathrm{η1}}{:=}{\mathrm{w3}}{}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ6}}{-}{2}{}{\mathrm{w1}}{}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ7}}{-}{2}{}{\mathrm{w1}}{}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ6}}{+}{\mathrm{w2}}{}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ7}}{+}{\mathrm{w3}}{}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ7}}$ (2.12)
 algW > $\mathrm{LD2}≔\mathrm{Deformation}\left(\mathrm{alg},\left[\mathrm{η1}\right],\mathrm{κ},\mathrm{algD2}\right)$
 ${\mathrm{LD2}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{κ}}{}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e7}}\right]{=}{2}{}{\mathrm{κ}}{}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{2}{}{\mathrm{κ}}{}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{κ}}{}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{κ}}{}{\mathrm{e3}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e5}}\right]$ (2.13)
 alg > $\mathrm{DGsetup}\left(\mathrm{LD2}\right)$
 ${\mathrm{Lie algebra: algD2}}$ (2.14)

This time the linear deformation defined by ${\mathrm{η}}_{1}$ does not satisfied the Jacobi identity.

 algD2 > $\mathrm{Query}\left("Jacobi"\right)$
 ${\mathrm{false}}$ (2.15)

To continue, we calculate the quadratic deformation. For this, we need the Massey product of ${\mathrm{η}}_{1}$with itself.

 algW > $\mathrm{ζ2}≔\mathrm{MasseyProduct}\left(\mathrm{η1},\mathrm{η1}\right)$
 ${\mathrm{ζ2}}{:=}{3}{}{\mathrm{w3}}{}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ6}}{}{\bigwedge }{}{\mathrm{θ7}}{+}{6}{}{\mathrm{w1}}{}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ6}}{}{\bigwedge }{}{\mathrm{θ7}}$ (2.16)

Next we use the command CohomologyDecomposition to determine if the Massey product  is exact.

 algW > $C,\mathrm{η2}≔\mathrm{CohomologyDecomposition}\left(-\mathrm{ζ2},\left[\right]\right)$
 ${C}{,}{\mathrm{η2}}{:=}{0}{}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ3}}{,}{3}{}{\mathrm{w4}}{}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ7}}{+}{6}{}{\mathrm{w5}}{}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ7}}{-}{3}{}{\mathrm{w8}}{}{\mathrm{θ5}}{}{\bigwedge }{}{\mathrm{θ7}}$ (2.17)

The 3-form ${\mathrm{ζ}}_{2}$ is exact. The second order deformation term is given by

 algW > $\left(\mathrm{ExteriorDerivative}\left(\mathrm{η2}\right)\right)&plus\mathrm{ζ2}$
 ${0}{}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ3}}$ (2.18)

We find the second order deformation to the original Lie algebra.

 Alg1 > $\mathrm{LD22}≔\mathrm{Deformation}\left(\mathrm{alg},\left[\mathrm{η1},\mathrm{η2}\right],\mathrm{κ},\mathrm{algD22}\right)$
 ${\mathrm{LD22}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{κ}}{}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e7}}\right]{=}{2}{}{\mathrm{κ}}{}{\mathrm{e1}}{-}{3}{}{{\mathrm{κ}}}^{{2}}{}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{2}{}{\mathrm{κ}}{}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{κ}}{}{\mathrm{e2}}{-}{6}{}{{\mathrm{κ}}}^{{2}}{}{\mathrm{e5}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{κ}}{}{\mathrm{e3}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e2}}{+}{3}{}{{\mathrm{κ}}}^{{2}}{}{\mathrm{e8}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e5}}\right]$ (2.19)
 alg > $\mathrm{DGsetup}\left(\mathrm{LD22}\right)$
 ${\mathrm{Lie algebra: algD22}}$ (2.20)
 alg > $\mathrm{Query}\left("Jacobi"\right)$
 ${\mathrm{false}}$ (2.21)

The second order deformation also fails to satisfy the Jacobi identity so we repeat the previous steps to find the third order deformation.

 algD22 > $\mathrm{ζ3}≔\left(\mathrm{MasseyProduct}\left(\mathrm{η1},\mathrm{η2}\right)\right)&plus\left(\mathrm{MasseyProduct}\left(\mathrm{η2},\mathrm{η1}\right)\right)$
 ${\mathrm{ζ3}}{:=}{-}{6}{}{\mathrm{w4}}{}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ6}}{}{\bigwedge }{}{\mathrm{θ7}}$ (2.22)
 algD22 > $C,\mathrm{η3}≔\mathrm{CohomologyDecomposition}\left(-\mathrm{ζ3},\left[\right]\right)$
 ${C}{,}{\mathrm{η3}}{:=}{0}{}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ3}}{,}{6}{}{\mathrm{w8}}{}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ7}}$ (2.23)

The next Massey products are zero. This means that the third order deformation is a Lie algebra.

 algW > $\mathrm{MasseyProduct}\left(\mathrm{η1},\mathrm{η3}\right)$
 ${0}{}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ3}}$ (2.24)
 algW > $\mathrm{MasseyProduct}\left(\mathrm{η2},\mathrm{η2}\right)$
 ${0}{}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ3}}$ (2.25)
 Alg1 > $\mathrm{LD23}≔\mathrm{Deformation}\left(\mathrm{alg},\left[\mathrm{η1},\mathrm{η2},\mathrm{η3}\right],\mathrm{κ},\mathrm{algD23}\right)$
 ${\mathrm{LD23}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{κ}}{}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e7}}\right]{=}{2}{}{\mathrm{κ}}{}{\mathrm{e1}}{-}{3}{}{{\mathrm{κ}}}^{{2}}{}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{2}{}{\mathrm{κ}}{}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{κ}}{}{\mathrm{e2}}{-}{6}{}{{\mathrm{κ}}}^{{2}}{}{\mathrm{e5}}{-}{6}{}{{\mathrm{κ}}}^{{3}}{}{\mathrm{e8}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{κ}}{}{\mathrm{e3}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e2}}{+}{3}{}{{\mathrm{κ}}}^{{2}}{}{\mathrm{e8}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e5}}\right]$ (2.26)
 alg > $\mathrm{DGsetup}\left(\mathrm{LD23}\right)$
 ${\mathrm{Lie algebra: algD23}}$ (2.27)
 algW > $\mathrm{Query}\left("Jacobi"\right)$
 ${\mathrm{true}}$ (2.28)

Example 3.

Here we using the calling sequence with the keyword argument parameters to find the most general linear deformation that can be constructed from the first 4 cohomology classes in $\mathrm{H2}$.

 algD23 > $\mathrm{η1}≔\mathrm{evalDG}\left(\mathrm{a1}{\mathrm{H2}}_{1}+\mathrm{a2}{\mathrm{H2}}_{2}+\mathrm{a3}{\mathrm{H2}}_{3}+\mathrm{a4}{\mathrm{H2}}_{4}\right)$
 ${\mathrm{η1}}{:=}\left({\mathrm{a3}}{}{\mathrm{w3}}{+}{\mathrm{a2}}{}{\mathrm{w2}}\right){}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ6}}{-}\left({-}{\mathrm{a4}}{}{\mathrm{w3}}{+}{2}{}{\mathrm{a3}}{}{\mathrm{w1}}\right){}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ7}}{-}\left({-}{\mathrm{a4}}{}{\mathrm{w3}}{+}{2}{}{\mathrm{a3}}{}{\mathrm{w1}}\right){}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ6}}{+}\left({-}{2}{}{\mathrm{a4}}{}{\mathrm{w1}}{+}{\mathrm{a3}}{}{\mathrm{w2}}\right){}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ7}}{+}\left({-}{2}{}{\mathrm{a2}}{}{\mathrm{w1}}{+}{\mathrm{a1}}{}{\mathrm{w2}}\right){}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ6}}{+}\left({\mathrm{a3}}{}{\mathrm{w3}}{+}{\mathrm{a2}}{}{\mathrm{w2}}\right){}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ7}}$ (2.29)
 algD24 > $\mathrm{TF},\mathrm{JacobiEq},\mathrm{JacobiSoln},\mathrm{LD3}≔\mathrm{Deformation}\left(\mathrm{alg},\left[\mathrm{η1}\right],\mathrm{κ},\mathrm{algD3},\mathrm{parameters}=\left\{\mathrm{a1},\mathrm{a2},\mathrm{a3},\mathrm{a4}\right\}\right)$
 ${\mathrm{TF}}{,}{\mathrm{JacobiEq}}{,}{\mathrm{JacobiSoln}}{,}{\mathrm{LD3}}{:=}{\mathrm{true}}{,}\left\{{0}{,}{3}{}{{\mathrm{κ}}}^{{2}}{}{{\mathrm{a3}}}^{{2}}{,}{6}{}{{\mathrm{κ}}}^{{2}}{}{{\mathrm{a3}}}^{{2}}{,}{-}{3}{}{{\mathrm{κ}}}^{{2}}{}{\mathrm{a2}}{}{\mathrm{a4}}{,}{3}{}{{\mathrm{κ}}}^{{2}}{}{\mathrm{a2}}{}{\mathrm{a4}}{,}{-}{{\mathrm{κ}}}^{{2}}{}{\mathrm{a1}}{}{\mathrm{a4}}{+}{4}{}{{\mathrm{κ}}}^{{2}}{}{\mathrm{a2}}{}{\mathrm{a3}}{,}{8}{}{{\mathrm{κ}}}^{{2}}{}{\mathrm{a2}}{}{\mathrm{a3}}{-}{2}{}{{\mathrm{κ}}}^{{2}}{}{\mathrm{a1}}{}{\mathrm{a4}}\right\}{,}\left[\left\{{\mathrm{a1}}{=}{\mathrm{a1}}{,}{\mathrm{a2}}{=}{\mathrm{a2}}{,}{\mathrm{a3}}{=}{0}{,}{\mathrm{a4}}{=}{0}\right\}{,}\left\{{\mathrm{a1}}{=}{0}{,}{\mathrm{a2}}{=}{0}{,}{\mathrm{a3}}{=}{0}{,}{\mathrm{a4}}{=}{\mathrm{a4}}\right\}\right]{,}\left[\left[\left[{\mathrm{e1}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{κ}}{}{\mathrm{a2}}{}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e6}}\right]{=}{2}{}{\mathrm{κ}}{}{\mathrm{a2}}{}{\mathrm{e1}}{-}{\mathrm{κ}}{}{\mathrm{a1}}{}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{κ}}{}{\mathrm{a2}}{}{\mathrm{e2}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e5}}\right]{,}\left[\left[{\mathrm{e1}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{κ}}{}{\mathrm{a4}}{}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{κ}}{}{\mathrm{a4}}{}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e7}}\right]{=}{2}{}{\mathrm{κ}}{}{\mathrm{a4}}{}{\mathrm{e1}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e5}}\right]\right]$ (2.30)

We therefore have two possibilities.The first is

 algD24 > $\mathrm{DGsetup}\left({\mathrm{LD3}}_{1}\right)$
 ${\mathrm{Lie algebra: algD3_1}}$ (2.31)
 algD24 > $\mathrm{MultiplicationTable}\left("LieTable"\right)$ and the second is

 algD24 > $\mathrm{DGsetup}\left({\mathrm{LD3}}_{2}\right)$
 ${\mathrm{Lie algebra: algD3_2}}$ (2.32)
 algD24 > $\mathrm{MultiplicationTable}\left("LieTable"\right)$ 