Algorithms for algebras over global fields

The main results in this dissertation concern the computational complexity of structural decomposition problems in finite dimensional associative algebras over global fields (algebraic number fields and global function fields, i.e., function fields of plane algebraic curves over finite fields). Polynomial time algorithms for isolating the radical and finding the simple components of the semisimple part of an algebra over a global function field are presented. We propose a method for computing the dimension of minimal one-sided ideals of a simple algebra over a global field. The method is based on computing a maximal order in the algebra, a non-commutative analogue of the ring of algebraic integers in a number field. The algorithm makes oracle calls to factor integers in the number fields case. A generalization of the LLL basis reduction algorithm is used to demonstrate that computing a maximal order in an algebra isomorphic to the ring of 2x2 rational matrices is equivalent to finding an explicit isomorphism. We also present some applications, such as an efficient membership test in commutative matrix groups as well as a polynomial time method for computing dimensions of irreducible representations of finite groups over number fields. Some results are also valid in more general contexts. For example, a deterministic polynomial time method for finding a maximal toral subalgebra in a semisimple algebra is presented as an application of a method for computing a Cartan subalgebra in a Lie algebra.

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