3,788 research outputs found
Tame group actions on central simple algebras
We study actions of linear algebraic groups on finite-dimensional central
simple algebras. We describe the fixed algebra for a broad class of such
actions.Comment: 19 pages, LaTeX; slightly revised; final version will appear in
Journal of Algebr
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
The Use of Faraday Rotation Sign Maps as a Diagnostic for Helical Jet Magnetic Fields
We present maps of the sign of the Faraday Rotation measure [sign(RM)]
obtained from multi-frequency radio observations on the Very Long Baseline
Array (VLBA). Many of the Active Galactic Nuclei (AGN) considered have B-field
structures with a central "spine" of B-field orthogonal to the jet and/or a
longitudinal B-field near one or both edges of the jet. This structure can
plausibly be interpreted as being caused by a helical/toroidal jet magnetic
field. Faraday Rotation is a rotation of the plane of polarization that occurs
when the polarized radiation passes through a magnetized plasma. The sign of
the RM is determined by the direction of the line-of-sight B-Field in the
region causing the Faraday Rotation, and an ordered toroidal or helical
magnetic field associated with an AGN jet will thus produce a distinctive
bilateral distribution of positive and negative RMs across the jet. We present
and discuss sign(RM) maps and their possible interpretation regarding the
magnetic field geometries for several sources.Comment: From the proceedings of Beamed and Unbeamed Gamma-Rays from Galaxies,
April 11-15, 2011, Muonio, Finland. 5 pages, 4 figure
A lower bound on the essential dimension of a connected linear group
Let G be a connected linear algebraic group defined over an algebraically
closed field k and H be a finite abelian subgroup of G whose order is prime to
char(k). We show that the essential dimension of G is bounded from below by
rank(H) - rank C_G(H)^0, where rank C_G(H)^0 denotes the rank of the maximal
torus in the centralizer C_G(H). This inequality, conjectured by J.-P. Serre,
generalizes previous results of Reichstein -- Youssin (where char(k) is assumed
to be 0 and C_G(H) to be finite) and Chernousov -- Serre (where H is assumed to
be a 2-group).Comment: 21 page
Group actions and invariants in algebras of generic matrices
We show that the fixed elements for the natural GL_m-action on the universal
division algebra UD(m,n) of m generic n x n matrices form a division subalgebra
of degree n, assuming n >= 3 and 2 <= m <= n^2 - 2. This allows us to describe
the asymptotic behavior of the dimension of the space of SL_m-invariant
homogeneous central polynomials p(X_1,...,X_m) for n x n matrices. Here the
base field is assumed to be of characteristic zero.Comment: 22 pages. Final version, to appear in Advances in Applied Mathematics
(Amitai Regev issue). Theorem 1.3 has been strengthene
- …