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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)
##### Keywords:
lattice; isomorphism; algorithm; order; number field
##### Software:
Nemo; Magma; Hecke
Full Text:
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