This package contains various algorithms related to finite dimensional nilpotent associative algebras. It also contains many group-theoretical functions related to the Modular Isomorphism Problem. We first give a brief introduction to finite dimensional nilpotent algebras and then an overview of the main algorithms.
Let A be an associative algebra of dimension d over a field F. Let {b_1, ..., b_d} be a basis for A. We identify the element x_1 b_1 + ... + x_d b_d of A with the element (x_1, ..., x_d) of F^d. The multiplication of A can then be described by a structure constants table: a 3-dimensional array with entries a_i,j,k ∈ F satisfying that
b_i b_j = \sum_{k=1}^d a_{i,j,k} b_k.
An associative algebra A is nilpotent if its power series terminates at the trivial ideal of A; that is
A > A^2 > \ldots > A^c > A^{c+1} = \{0\}
where A^j 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/A^2 is the rank of A. Note that A is generated by dim(A/A^2) 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 (w_1, ..., w_d) so that
A^j = \langle b_i \mid w_i \geq j \rangle.
Note that A A^j = A^j+1 for every j. Thus it is possible to choose all basis elements of weight at least 2 so that b_i = b_k b_l holds for some k and l, where b_k is of weight 1 and b_l is of weight w_i-1. This feature allows an effective description of A via a nilpotent structure constants table. This contains the structure constants a_i,j,k for all i with w_i = 1 and 1 ≤ j,k ≤ d. For i with w_i > 1 it either contains a description as b_i = b_k b_l or the structure constants a_i,j,k for 1 ≤ j,k ≤ d. It may also contain both or some partial overlap of this information.
Let A be a finite dimensional nilpotent associative algebra over a finite field. This package contains an implementation of the methods in [Eic08] which allow the determination of the automorphism group Aut(A) and a canonical form Can(A).
The automorphism group is given by generators and is 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 asks whether an isomorphism of algebras ℙ_p G ≅ ℙ_p H implies an isomorphism of groups G ≅ H for two p-groups G and H and ℙ_p the field with p elements. This problem was open for a long time until first counterexamples for the prime p=2 were found in [GMd22]. It remains open for odd primes and many other interesting classes of groups.
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 ℙ_p G and ℙ_p H are isomorphic, where G and H are finite p-groups. This algorithm has been implemented in C by Wursthorn and has been applied to the groups of order dividing 2^7 without finding a counterexample, see [BKRW99]. The implementation of Wursthorn appears to be lost, but is in any case not publicly available.
This package contains an implementation of the new algorithm described in [Eic08] 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 for G a finite p-group and FG is isomorphic to FH if and only if the radicals J(FG) and J(FH) are isomorphic. 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 of finite p-groups. Note that in this setting the Jacobson radical of the group algebra FG equals its augmentation ideal.
The methods of this package have been used to study the modular isomorphism problem for the groups of order dividing 3^6 and 2^8 ([Eic08]) and for the groups of order 2^9 ([EK11]). It was later used to study also groups of order 3^7 and 5^6 ([MM22]).
A property of a group G is called F-invariant if an isomorphism of F-algebras FG ≅ FH implies the same property for H. In the context of the Modular Isomorphism Problem, if G is a finite p-group, then an ℙ_p-invariant is simply called invariant. Many invariants of G are known and the package provides functions for them, as well as programs which make it easy to compare all the implemented invariants quickly for a given list of groups.
It also remains open whether replacing the field ℙ_p in the Modular Isomorphism Problem with a bigger field of characteristic p will change the outcome of the problem for a given pair of groups. The package includes several functions which make it possible to investigate this question by applying the algorithm for the same groups varying the field.
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, i.e. the algebra A/A^c+1. See [Eic11] for details on the underlying algorithm.
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) / \langle \langle w^n \mid w \in F(d,F) \rangle \rangle
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 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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