[ModIsom] 1 Introduction

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# 1 Introduction

This package contains various algorithms related to finite dimensional nilpotent associative algebras. We first give a brief introduction to these algebras and then an overview of the main algorithms.

Associative algebras and nilpotency

Let A be an associative algebra of dimension d over a field F. Let {b1, …, bd} be a basis for A. We identify the element x1 b1 + …+ xd bd of A with the element (x1, …, xd) of Fd. The multiplication of A can then be described by a structure constants table: a 3-dimensional array with entries ai,j,k ∈ F satisfying that

bi bj = | d ∑ k=1 | ai,j,k bk·

An associative algebra A is nilpotent if its power series terminates at the trivial ideal of A; that is

A > A2 > … > Ac > Ac+1 = {0}

where Aj is the ideal of A generated by all products of length at least j. The length c of the power series is also called the class of A and the dimension of A/A2 is the rank of A. Note that A is generated by dim(A/A2) elements. Clearly, A does not contain a multiplicative identity.

For computational purposes we describe a nilpotent associative algebra by a weighted basis and a description of the corresponding structure constants table. A basis of a nilpotent associative algebra A is weighted if there is a sequence of weights (w1, …, wd) so that

Aj = 〈bi | wi ≥ j 〉·

Note that A Aj = Aj+1 for every j. Thus it is possible to choose all basis elements of weight at least 2 so that bi = bk bl holds for some k and l, where bk is of weight 1 and bl is of weight wi−1. This feature allows an effective description of A via a nilpotent structure constants table. This contains the structure constants ai,j,k for all i with wi = 1 and 1 ≤ j,k ≤ d. For i with wi > 1 it either contains a description as bi = bk bl or the structure constants ai,j,k for 1 ≤ j,k ≤ d. It may also contain both or some partial overlap of these informations.

Isomorphisms and Automorphisms

Let A be a finite dimensional nilpotent associative algebra over a finite field. This package contains an implementation of the methods in Eic07 which allow the determination of the automorphism group Aut(A) and a canonical form Can(A).

The automorphism group is given by generators and it represented as a subgroup of GL(dim(A), F). Also the order of Aut(A) is available.

A canonical form Can(A) for A is a nilpotent structure constants table for A which is unique for the isomorphism type of A; that is, two algebras A and B are isomorphic if and only if Can(A) = Can(B) holds. Hence the canonical form can be used to solve the isomorphism problem.

The modular isomorphism problem

The modular isomorphism problem asks whether FG ≅ FH implies that G ≅ H for two p-groups G and H and F the field with p elements. This problem is still open, despite various efforts towards proving the claim or finding counterexamples to it.

Computational approaches have been used to investigate the modular isomorphism problem. Based on an algorithm by Roggenkamp and Scott RS93, Wursthorn Wur93 described an algorithm for checking the modular isomorphism problem; that is, he described an algorithm for checking whether two modular group algebras FG and FH are isomorphic. This algorithm has been implemented in C by Wursthorn and has been used applied to the groups of order dividing 27 without finding a counterexample, see BKRW99.

This package contains an implementation of the new algorithm described in Eic07 for checking isomorphism of modular group algebras. It is based on the fact that the Jacobson radical J(FG) is nilpotent if FG is a modular group algebra. Hence the automorphism group and canonical form algorithm of this package apply and can be used to solve the isomorphism problem for modular group algebras.

The methods of this package have been used to check the modular isomorphism problem for the groups of order dividing 36 and 28 (Eic07) and for the groups of order 29 (EKo11).

A nilpotent quotient algorithm

Given a finitely presented associative algebra A over an arbitrary field F, this package contains an algorithm to determine a nilpotent structure constants table for the class-c nilpotent quotient of A. See Eic11 for details on the underlying algorithm.

Kurosh Algebras

Let F(d,F) denote the free non-unital associative algebra on d generators over the field F. Then

A(d,n,F) = F(d,F) / 〈〈wn | w ∈ F(d,F) 〉〉

is the Kurosh Algebra on d generators of exponent n over the field F. Kurosh Algebras can be considered as an algebra-theoretic analogue to Burnside groups.

This package contains a method that allows to determine A(d,n,F) for given d, n, F. This can also be used to determine A(d,n,F) for all fields of a given characteristic. We refer to Eic11 for details on the algorithms.

This package also contains a database of Kurosh Algebras that have been determined with the methods of this package.

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