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Computing isomorphisms between lattices. (English) Zbl 1452.11134
Let \(K\) be a number field with ring of integers \(O_K\). Let \(A\) be a finite-dimensional semisimple \(K\)-algebra, and \(A = A_1 \oplus \cdots \oplus A_r\) be the decomposition of \(A\) into indecomposable two-sided ideals, and denote by \(K_i\) the center of the simple algebra \(A_i\). We consider the following two hypotheses:
(H1) For each \(i\), we can compute an explicit isomorphism \(A_i \cong Mat_{n_i\times n_i}(D_i)\) of \(K\)-algebras, where \(D_i\) is a skew field with center \(K_i\).
(H2) For every maximal \(O_K\)-order \(\Delta_i\) in \(D_i\) we can solve the principal ideal problem for fractional left \(\Delta_i\)-ideals, and \(\Delta_i\) has the locally free cancellation property.
Let \(\Lambda\) be an \(O_K\)-order in \(A\). Recall that a \(\Lambda\)-lattice is a (left) \(\Lambda\)-module that is finitely generated and torsion-free over \(O_K\). In this paper, it is proved, under the above hypotheses, that there exists an algorithm that for two given \(\Lambda\)-lattices \(X\) and \(Y\) either computes an isomorphism \(X\rightarrow Y\) or determines that \(X\) and \(Y\) are not isomorphic. The algorithm is implemented in the algebraic computational package Magma for \(A = Q[G]\), \(\Lambda = Z[G]\), and \(\Lambda\)-lattices \(X\) and \(Y\) contained in \(Q[G]\), where \(G\) is a finite group satisfying certain hypotheses. The implementation can decide whether two \(Z[G]\)-lattices contained in \(Q[G]\) are isomorphic and, if so, gives an explicit isomorphism. Experimental results are discussed.
MSC:
11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers
11Y40 Algebraic number theory computations
16Z05 Computational aspects of associative rings (general theory)
Software:
Nemo; Magma; Hecke
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