[Paper Review] Computing 2-Dimensional Algebras: Crossed Modules and Cat1-Algebras

Z. Arvasi, Alper Odabaş|arXiv (Cornell University)|Nov 26, 2013

Algebraic structures and combinatorial models8 references3 citations

TL;DR

This paper presents a GAP implementation for computing 2-dimensional algebraic structures, specifically crossed modules and cat1-algebras of commutative algebras over a commutative ring. It establishes the equivalence between the categories XModAlg and Cat1-Alg, providing algorithms and functions to compute morphisms, endomorphisms, and idempotent endomorphisms, with a comprehensive table of cat1-structures on group algebras over small finite fields.

ABSTRACT

In this paper, we described the GAP implementation of crossed modules of commutative algebras and cat$^{1}$-algebras and their equivalence.

Motivation & Objective

• To develop computational tools in GAP for 2-dimensional algebraic structures, particularly crossed modules and cat1-algebras.
• To establish the categorical equivalence between XModAlg (crossed modules of algebras) and Cat1-Alg (cat1-algebras).
• To implement algorithms for computing morphisms, endomorphisms, and idempotent endomorphisms in the context of group algebras.
• To provide a systematic table of cat1-structures on group algebras over finite fields of order 2, 3, and 4 for groups of order up to 12.
• To extend computational group theory techniques to the setting of commutative algebras, despite limitations in native GAP support for semidirect products and multiplier algebras.

Proposed method

• Adapted the group-theoretic XMod package for GAP4 to the commutative algebra setting, defining k-algebra morphisms and actions satisfying XModAlg1 and XModAlg2 axioms.
• Defined crossed modules as k-algebra morphisms ∂: S → R with an R-action on S satisfying ∂(r·s) = r∂(s) and ∂(s)·s′ = ss′ for all r ∈ R, s, s′ ∈ S.
• Constructed cat1-algebras from crossed modules via the functorial correspondence: a cat1-algebra consists of a k-algebra A with two endomorphisms s, t: A → A satisfying s∘t = s and t∘s = t.
• Implemented functions in GAP to compute: XModAlgByIdeal, Cat1AlgByXModAlg, XModAlgByCat1Alg, and Cat1AlgByEndomorphisms.
• Analyzed endomorphisms and idempotent endomorphisms of group algebras FG for finite fields F and groups G of order ≤12.
• Generated a comprehensive table of cat1-structures by enumerating all idempotent endomorphism pairs and counting valid cat1-algebra configurations.

Experimental results

Research questions

• RQ1 How can the categorical equivalence between crossed modules of commutative algebras and cat1-algebras be algorithmically realized in computational algebra systems?
• RQ2 What is the computational feasibility and performance of implementing crossed modules and cat1-algebras in GAP for finite group algebras?
• RQ3 How many distinct cat1-structures exist on group algebras FG for small finite fields F and groups G of order ≤12?
• RQ4 What are the structural properties of endomorphisms and idempotent endomorphisms in group algebras over GF(2), GF(3), and GF(4)?
• RQ5 Can the GAP framework be extended to support key algebraic constructions like multiplier algebras and semidirect products in the context of commutative algebras?

Key findings

• The paper successfully implemented the equivalence between XModAlg and Cat1-Alg in GAP, with functions Cat1Alg and XModAlg enabling bidirectional conversion between structures.
• For the group algebra FG with G = C2 and F = GF(4), the number of cat1-structures is 2, with 5 endomorphisms and 3 idempotent endomorphisms.
• In the case of G = C2 × C2 and F = GF(4), the group algebra FG has 81 cat1-structures, arising from 4097 endomorphisms and 83 idempotent endomorphisms.
• The largest number of cat1-structures found was 25 for G = C2 × C2 and F = GF(3), with 625 endomorphisms and 104 idempotent endomorphisms.
• For G = C2 × C2 × C2 and F = GF(2), the group algebra FG has 87 distinct cat1-structures, with over 2.6 million endomorphisms and 723 idempotent endomorphisms.
• The implementation correctly verifies isomorphisms: converting a crossed module to a cat1-algebra and back recovers the original structure, as confirmed by GAP function equality checks.