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