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2 Descendants of nilpotent associative algebras
 2.1 Immediate descendants
 2.2 Classifying nilpotent associative algebras by dimension
 2.3 Enumerating nilpotent associative algebras

2 Descendants of nilpotent associative algebras

2.1 Immediate descendants

Let F be a finite field and let A and B be two nilpotent associative F-algebras. Then B is called an immediate descendant of A if A is isomorphic to B/B^cl(B). The difference dim(B)-dim(A) is called the stepsize of the descendant. A nilpotent associative F-algebra is called capable, if it has descendants. In [EM17] an algorithm is described to calculate these immediate descendants. This algorithm and the capability check is implemented in the following functions.

2.1-1 ImmediateDescendantsOfCanoForm
‣ ImmediateDescendantsOfCanoForm( C, s )( operation )

Given a canonical form C this function returns the immediate descendants of C with stepsize s.

2.1-2 ImmediateDescendantsOfCanoForm
‣ ImmediateDescendantsOfCanoForm( C, slist )( operation )

Given a canonical form C this function returns the immediate descendants of C with stepsizes in slist.

2.1-3 ImmediateDescendantsOfCanoForm
‣ ImmediateDescendantsOfCanoForm( C )( operation )

Given a canonical form C this function returns the immediate descendants of C of all stepsizes.

2.1-4 IsCapableCanoForm
‣ IsCapableCanoForm( C )( property )

This returns whether the given canonical form C is capable, i.e. whether it has descendants.

2.2 Classifying nilpotent associative algebras by dimension

The descendant algorithm can be used for a classification of nilpotent associative F-algebras by dimension. This is used in the following functions.

2.2-1 CanoFormsByDimAndGens
‣ CanoFormsByDimAndGens( F, d, n )( operation )

Given a finite field F, a generator number d and a dimension n, this function calculates canonical forms for the d-generator nilpotent associative F-algebras of dimension n.

2.2-2 CanoFormsByDim
‣ CanoFormsByDim( F, n )( operation )

Given a finite field F and a dimension n, this function calculates canonical forms for the nilpotent associative F-algebras of dimension n.

2.3 Enumerating nilpotent associative algebras

Sometimes it will be enough to enumerate the number of immediate descendants or the number of nilpotent associative algebras of a given dimension. This is implemented in the following functions using orbit counting methods. However, note that the enumeration of n-dimensional algebras still requires the computation of the (n-1)-dimensional algebras.

2.3-1 NumberOfImmediateDescendantsOfCanoForm
‣ NumberOfImmediateDescendantsOfCanoForm( C, s )( operation )

Given a canonical form C this returns the number of immediate descendants of C with stepsize s.

2.3-2 NumberOfImmediateDescendantsOfCanoForm
‣ NumberOfImmediateDescendantsOfCanoForm( C, slist )( operation )

Given a canonical form C this returns the number of immediate descendants of C with stepsizes in slist.

2.3-3 NumberOfImmediateDescendantsOfCanoForm
‣ NumberOfImmediateDescendantsOfCanoForm( C )( operation )

Given a canonical form C this returns the number of immediate descendants of C of all stepsizes.

2.3-4 NumberOfNAAlgsByDimAndGens
‣ NumberOfNAAlgsByDimAndGens( F, d, n )( operation )

Given a finite field F, a generator number d and a dimension n, this function returns the number of d-generator nilpotent associative F-algebras of dimension n.

2.3-5 NumberOfNAAlgsByDim
‣ NumberOfNAAlgsByDim( F, n )( operation )

Given a finite field F and a dimension n, this function returns the number of nilpotent associative F-algebras of dimension n.

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