1,407 research outputs found
Symmetric spaces and Lie triple systems in numerical analysis of differential equations
A remarkable number of different numerical algorithms can be understood and
analyzed using the concepts of symmetric spaces and Lie triple systems, which
are well known in differential geometry from the study of spaces of constant
curvature and their tangents. This theory can be used to unify a range of
different topics, such as polar-type matrix decompositions, splitting methods
for computation of the matrix exponential, composition of selfadjoint numerical
integrators and dynamical systems with symmetries and reversing symmetries. The
thread of this paper is the following: involutive automorphisms on groups
induce a factorization at a group level, and a splitting at the algebra level.
In this paper we will give an introduction to the mathematical theory behind
these constructions, and review recent results. Furthermore, we present a new
Yoshida-like technique, for self-adjoint numerical schemes, that allows to
increase the order of preservation of symmetries by two units. Since all the
time-steps are positive, the technique is particularly suited to stiff
problems, where a negative time-step can cause instabilities
Generating the Johnson filtration
For k >= 1, let Torelli_g^1(k) be the k-th term in the Johnson filtration of
the mapping class group of a genus g surface with one boundary component. We
prove that for all k, there exists some G_k >= 0 such that Torelli_g^1(k) is
generated by elements which are supported on subsurfaces whose genus is at most
G_k. We also prove similar theorems for the Johnson filtration of Aut(F_n) and
for certain mod-p analogues of the Johnson filtrations of both the mapping
class group and of Aut(F_n). The main tools used in the proofs are the related
theories of FI-modules (due to the first author together with Ellenberg and
Farb) and central stability (due to the second author), both of which concern
the representation theory of the symmetric groups over Z.Comment: 32 pages; v2: paper reorganized. Final version, to appear in Geometry
and Topolog
Homogeneous projective bundles over abelian varieties
We consider those projective bundles (or Brauer-Severi varieties) over an
abelian variety that are homogeneous, i.e., invariant under translation. We
describe the structure of these bundles in terms of projective representations
of commutative algebraic groups; the irreducible bundles correspond to
Heisenberg groups and their standard representations. Our results extend those
of Mukai on semi-homogeneous vector bundles, and yield a geometric view of the
Brauer group of abelian varieties.Comment: Final version, accepted for publication in Algebra and Number Theory
Journal; 37 pages. This is a slightly shortened version of v3: Section 6 has
been suppressed as well as the proofs of Propositions 4.1 and 4.2; Section 4
has been relegated to the very en
Commensurated subgroups of arithmetic groups, totally disconnected groups and adelic rigidity
Investigations into and around a 30-year old conjecture of Gregory Margulis
and Robert Zimmer on the commensurated subgroups of S-arithmetic groups.Comment: 50 page
Group actions on central simple algebras: a geometric approach
We study actions of linear algebraic groups on central simple algebras using
algebro-geometric techniques. Suppose an algebraic group G acts on a central
simple algebra A of degree n. We are interested in questions of the following
type: (a) Do the G-fixed elements form a central simple subalgebra of A of
degree n? (b) Does A have a G-invariant maximal subfield? (c) Does A have a
splitting field with a G-action, extending the G-action on the center of A?
Somewhat surprisingly, we find that under mild assumptions on A and the
actions, one can answer these questions by using techniques from birational
invariant theory (i.e., the study of group actions on algebraic varieties, up
to equivariant birational isomorphisms). In fact, group actions on central
simple algebras turn out to be related to some of the central problems in
birational invariant theory, such as the existence of sections, stabilizers in
general position, affine models, etc. In this paper we explain these
connections and explore them to give partial answers to questions (a)-(c).Comment: 33 pages. Final version, to appear in Journal of Algebra. Includes a
short new section on Brauer-Severi varietie
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