1,119 research outputs found
Splitting full matrix algebras over algebraic number fields
Let K be an algebraic number field of degree d and discriminant D over Q. Let
A be an associative algebra over K given by structure constants such that A is
isomorphic to the algebra M_n(K) of n by n matrices over K for some positive
integer n. Suppose that d, n and D are bounded. Then an isomorphism of A with
M_n(K) can be constructed by a polynomial time ff-algorithm. (An ff-algorithm
is a deterministic procedure which is allowed to call oracles for factoring
integers and factoring univariate polynomials over finite fields.)
As a consequence, we obtain a polynomial time ff-algorithm to compute
isomorphisms of central simple algebras of bounded degree over K.Comment: 15 pages; Theorem 2 and Lemma 8 correcte
Locally Equivalent Correspondences
Given a pair of number fields with isomorphic rings of adeles, we construct
bijections between objects associated to the pair. For instance we construct an
isomorphism of Brauer groups that commutes with restriction. We additionally
construct bijections between central simple algebras, maximal orders, various
Galois cohomology sets, and commensurability classes of arithmetic lattices in
simple, inner algebraic groups. We show that under certain conditions, lattices
corresponding to one another under our bijections have the same covolume and
pro-congruence completion. We also make effective a finiteness result of Prasad
and Rapinchuk.Comment: Final Version. To appear in Ann. Inst. Fourie
Transitive Lie algebras of vector fields---an overview
This overview paper is intended as a quick introduction to Lie algebras of
vector fields. Originally introduced in the late 19th century by Sophus Lie to
capture symmetries of ordinary differential equations, these algebras, or
infinitesimal groups, are a recurring theme in 20th-century research on Lie
algebras. I will focus on so-called transitive or even primitive Lie algebras,
and explain their theory due to Lie, Morozov, Dynkin, Guillemin, Sternberg,
Blattner, and others. This paper gives just one, subjective overview of the
subject, without trying to be exhaustive.Comment: 20 pages, written after the Oberwolfach mini-workshop "Algebraic and
Analytic Techniques for Polynomial Vector Fields", December 2010 2nd version,
some minor typo's corrected and some references adde
A Lie Algebra Method for Rational Parametrization of Severi-Brauer Surfaces
It is well-known that a Severi-Brauer surface has a rational point if and
only if it is isomorphic to the projective plane. Given a Severi-Brauer
surface, we study the problem to decide whether such an isomorphism to the
projective plane, or such a rational point, does exist; and to construct such
an isomorphism or such a point in the affirmative case. We give an algorithm
using Lie algebra techniques. The algorithm has been implemented in Magma.Comment: 16 pages some minor revision
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