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

Actions of Algebraic Groups on the Spectrum of Rational Ideals, II

AbstractWe study rational actions of a linear algebraic groupGon an algebraV, and the induced actions on Rat(V), the spectrum of rational ideals ofV(a subset of Spec(V) which often includes all primitive ideals). This work extends results of Moeglin and Rentschler to prime characteristic, often also relaxing their finiteness assumptions onV. In particular, we relate properties of a rational idealJand itsorb, the ideal (J:G)=⋂γ∈Gγ(J). The rational ideals ofVcontaining the orb ofJare precisely those in the Zariski-closureXof the orbit ofJin Rat(V). TheG-stratumofJconsists of all rational ideals inXwhose orbit is dense inX(i.e., whose orb is equal to the orb ofJ). We show that theG-stratum of a rational ideal consists of exactly oneG-orbit, and that rational ideals are maximal in their strata in a strong sense. These results are useful for studying prime and primitive spectra of certain algebras, cf. recent work by Goodearl and Letzter. We further show that the orbit ofJis open in its closure in Rat(V), provided thatJis locally closed. Among other results, we show that the semiprime ideal (J:G) is Goldie, and we relate the uniform and Gelfand–Kirillov dimensions ofV/JandV/(J:G)

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