Improved algorithms for splitting full matrix algebras

TL;DR

This paper presents improved polynomial-time algorithms for explicitly constructing isomorphisms between associative algebras and full matrix algebras over number fields, with specific enhancements for certain small degrees and base fields.

Contribution

The authors simplify and enhance previous algorithms for algebra isomorphism construction, focusing on cases with small matrix sizes and specific base fields, leveraging lattice tensor product results.

Findings

Algorithms are now more efficient for n ≤ 43 over Q.

Improved methods for n=2 over quadratic imaginary fields.

Reduction in computational complexity for specific algebra cases.

Abstract

Let $\K$ be an algebraic number field of degree $d$ and discriminant $\Delta$ over $\Q$. Let $\A$ be an associative algebra over $\K$ given by structure constants such that $\A\cong M_n(\K)$ holds for some positive integer $n$. Suppose that $d$, $n$ and $|\Delta|$ are bounded. In a previous paper a polynomial time ff-algorithm was given to construct explicitly an isomorphism $\A \rightarrow M_n(\K)$. Here we simplify and improve this algorithm in the cases $n\leq 43$, $\K=\Q$, and $n=2$, with $\K=\Q(\sqrt{-1})$ or $\K=\Q(\sqrt{-3})$. The improvements are based on work by Y. Kitaoka and R. Coulangeon on tensor products of lattices.