`> SophusTest` ( ) | ( function ) |

Tests **Sophus** functions, returns true if it finds no mistakes, and returns false otherwise. May take a couple of minutes to complete.

`> IsLieNilpotentOverFp` ( L ) | ( property ) |

Returns true if `L` is a nilpotent Lie algebra and its underlying field is a finite prime field.

`> MinimalGeneratorNumber` ( L ) | ( attribute ) |

Computes the minimal number of generators for $L$, which is the dimension of $L/L'$.

`> AbelianLieAlgebra` ( F, d ) | ( function ) |

Returns the Abelian Lie algebra with dimension $d$ over the field `F`.

`> NilpotentBasis` ( L ) | ( attribute ) |

Computes a nilpotent basis for $L$. Nilpotent bases are defined in Section **1.**.

`> LieNBWeights` ( B ) | ( attribute ) |

Every element of the nilpotent basis $B$ has a weight; See Section **1.**. This function returns the list of these weights.

`> LieNBDefinitions` ( B ) | ( attribute ) |

This function returns a list. The `i`-th element of this list is 0 if `B[i]` has weight 1. Otherwise the `i`-th element is `[k,l]` if the definition of `B[i]` is `[B[k],B[l]]`. See Section **1.**.

`> IsNilpotentBasis` ( B ) | ( property ) |

Returns `true`

if the basis `B` of a Lie algebra was computed with the function `NilpotentBasis`

; `false`

otherwise.

`> IsLieAlgebraWithNB` ( L ) | ( property ) |

Returns `true`

if a nilpotent basis for `L` has already been computed using the function `NilpotentBasis`

; `false`

otherwise.

`> LieCover` ( L ) | ( attribute ) |

Computes the cover for the nilpotent Lie algebra $L$ as defined in Section **1.**.

`> CoverHomomorphism` ( C ) | ( attribute ) |

The nilpotent Lie algebra `C` was obtained from a nilpotent Lie algebra `L` using the `LieCover( L )` function call. This function returns the natural homomorphism from `C` onto `L`.

`> CoverOf` ( C ) | ( attribute ) |

The nilpotent Lie algebra `C` was obtained from a nilpotent Lie algebra `L` using the `LieCover( L )` function call. This function returns `L`.

`> IsLieCover` ( C ) | ( property ) |

Returns `true`

if the Lie algebra `C` was obtained as the Lie cover of another Lie algebra `L` using the `LieCover( L )` function call.

`> LieMultiplicator` ( C ) | ( attribute ) |

The nilpotent Lie algebra `C` was obtained from a nilpotent Lie algebra `L` using the `LieCover( L )` function call. This function returns the central ideal of `C` which is the multiplicator of `L`; see Section **1.**.

`> LieNucleus` ( C ) | ( attribute ) |

The nilpotent Lie algebra `C` was obtained from a nilpotent Lie algebra `L` using the `LieCover( L )` function call. This function returns the central ideal of `C` which is the nucleus of `L`; see Section **1.**.

We define a special class of automorphisms for our work.

`> NilpotentLieAutomorphism` ( L, gens, imgs ) | ( method ) |

`L` is a nilpotent Lie algebra, `gens` is a generating set, and `imgs` is a subset of `L` with the same length as `gens`. Returns the automorphism of `L` which maps the element of `gens` to the elements of `imgs`. It is the responsibility of the user to make sure that the arguments are given so that the automorphism exists. These automorphisms can be compared, multiplied using the `*` sign, and the inverse of such an automorphism can also be computed in the usual manner.

`> IdentityNilpotentLieAutomorphism` ( L ) | ( method ) |

`L` is a nilpotent Lie algebra; returns the identity automorphism of $L$.

`> IsNilpotentLieAutomorphism` ( A ) | ( property ) |

Returns `true`

if `A` was obtained using a `NilpotentLieAutomorphism` or an `IdentityNilpotentLieAutomorphism` function call.

`> AutomorphismGroup` ( L ) | ( method ) |

`L` is a nilpotent Lie algebra; returns the automorphism group of `L` as a group generated by **GAP** algebra automorphisms. The automorphism group is computed as explained in [S].

`> AutomorphismGroupNilpotentLieAlgebra` ( L ) | ( method ) |

`L` is a nilpotent Lie algebra; returns the automorphism group of `L` in the internally used hybrid format. The automorphism group is computed as explained in [S]. The hybrid format, which is very similar to the one used in [EO], is a record that contains the following fields.

`glAutos`

: a set of automorphisms which together with`agAutos`

generate the automorphism group;`glOrder`

: an integer whose product with the numbers in`agOrder`

gives the size of the automorphism group;`agAutos`

: a polycyclic generating sequence for a soluble normal subgroup of the automorphism group;`agOrder`

: the relative orders corresponding to`agAutos`

;`liealg`

: The Lie algebra acted upon by the automorphisms.`size`

: the size of the automorphism group.`field`

: the underlying field of the Lie algebra.`prime`

: the characteristic of the underlying field.

We do not return an automorphism group in the standard form because we wish to distinguish between `agAutos`

and `glAutos`

; the latter act non-trivially on the derived quotient of L. This hybrid-group description of the automorphism group permits more efficient computations with it.

`> AreIsomorphicNilpotentLieAlgebras` ( L, K ) | ( method ) |

Returns `true`

if `L` and `K` are isomorphic; `false`

otherwise.

`> Descendants` ( L, step ) | ( method ) |

Returns the `step`

-step descendants of a nilpotent Lie algebra `L`.

`> DescendantsOfStep1OfAbelianLieAlgebra` ( L, step ) | ( method ) |

Returns the `1`

-step descendants of the abelian Lie algebra with dimension `d` defined over the field of `p` elements.

The package provides with a number of functions that can be used to store lists of Lie algebras. Here we document only the most important ones, see the source code `io.gi`

for the rest.

`> WriteLieAlgebraToString` ( L ) | ( function ) |

Returns a string that encodes the nilpotent Lie algebra `L`

`> ReadStringToNilpotentLieAlgebra` ( string, p, d ) | ( function ) |

Decodes `string` into a `d`-dimensional nilpotent Lie algebra defined over the field of `p` elements.

`> WriteLieAlgebraListToFile` ( list, name, file ) | ( function ) |

`list` is a list of nilpotent Lie algebras. Encodes each Lie algebra in `list` to a string. The list so obtained is written into `file`. The name of this list will be `name`.

`> SophusBuildManual` ( ) | ( function ) |

Builds Sophus manual.

`> SophusBuildManualHTML` ( ) | ( function ) |

Builds Sophus manual in html format.

generated by GAPDoc2HTML