Improved algorithms for splitting full matrix algebras

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