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3 Actions and classes of algebras whose isomorphism types can be enumerated
 3.1 Kronecker Action - Associative algebras
 3.2 More actions

3 Actions and classes of algebras whose isomorphism types can be enumerated

Here it is briefly introduced which actions are implemented and which classes of algebras can be enumerated.

When one follows [Eic08] one gets the relation between the desired class of algebras that shall be enumerated and the action that has to be used.

3.1 Kronecker Action - Associative algebras

In the case of class-2 nilpotent associative algebras of rank r one obtains that the general linear group mathrmGL(r,q) operates on the set of subspaces of the r^2 dimensional vector space M = F_q^r ⊗_F_q F_q^r via the Kronecker product.

To obtain the number of class-2 nilpotent algebras of rank r and dimension d over the field with q elements one hast to call the function NumberOfClassTwoAlgebras (2.2-1) wth the second argument Kronecker.

The algorithm uses some important properties arising from the action. Further, symmetries amongst the codimensions k are used:

N_{d,r}(q) = 0 \; \text{if} \; d \notin \{r+1, \ldots, r^2\} \; \text{and} \; N_{r+r^2,r}(q)=1.

\text{For}\;\text{all} \; k \in \{1, \ldots, r^2\} \; \text{holds} \; N_{r+k,r}(q) = N_{r+r^2,r}(q)=1.

A proof can be found in [EW17].

Especially the second equation is used for the output of NumberOfClassTwoAlgebras (2.2-1) when called with the second argument Kronecker: For a better reading, it will be returned a list of length ⌈fracr^22+1⌉, only. The latter half would be just symmetric to the printed functions.

3.2 More actions

In the future hopefully more actions will be implemented.

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