31 research outputs found
Is the function field of a reductive Lie algebra purely transcendental over the field of invariants for the adjoint action?
Let be a field of characteristic zero, let be a connected reductive
algebraic group over and let be its Lie algebra. Let ,
respectively, , be the field of -rational functions on ,
respectively, . The conjugation action of on itself induces
the adjoint action of on . We investigate the question
whether or not the field extensions and
are purely transcendental. We show that the
answer is the same for and ,
and reduce the problem to the case where is simple. For simple groups we
show that the answer is positive if is split of type or , and negative for groups of other types, except possibly . A
key ingredient in the proof of the negative result is a recent formula for the
unramified Brauer group of a homogeneous space with connected stabilizers.
As a byproduct of our investigation we give an affirmative answer to a
question of Grothendieck about the existence of a rational section of the
categorical quotient morphism for the conjugating action of on itself. The
results and methods of this paper have played an important part in recent A.
Premet's negative solution (arxiv:0907.2500) of the Gelfand--Kirillov
conjecture for finite-dimensional simple Lie algebras of every type, other than
, , and .Comment: Final version, 37 pages. To appear in Compositio Mathematica
On Modular Inverses of Cyclotomic Polynomials and the Magnitude of their Coefficients
Let p and r be two primes and n, m be two distinct divisors of pr. Consider
the n-th and m-th cyclotomic polynomials. In this paper, we present lower and
upper bounds for the coefficients of the inverse of one of them modulo the
other one. We mention an application to torus-based cryptography.Comment: 21 page
The Discriminant of an Algebraic Torus
For a torus T defined over a global field K, we revisit an analytic class
number formula obtained by Shyr in the 1970's as a generalization of
Dirichlet's class number formula. We prove a local-global presentation of the
quasi-discriminant of T, which enters into this formula, in terms of
cocharacters of T. This presentation can serve as a more natural definition of
this invariant.Comment: 17 page
Arithmetic Toric Varieties
We study toric varieties over a field k that split in a Galois extension K / k using Galois cohomology with coefficients in the toric automorphism group. Part of this Galois cohomology fits into an exact sequence induced by the presentation of the class group of the toric variety. This perspective helps to compute the Galois cohomology, particularly for cyclic Galois groups. We use Galois cohomology to classify k-forms of projective spaces when K / k is cyclic, and we also study k-forms of surfaces
Weakly commensurable S-arithmetic subgroups in almost simple algebraic groups of types B and C
Let G and G' be absolutely almost simple algebraic groups of types B and C
respectively, of rank at least 3, and defined over a number field K. We
determine when G and G' have the same isomorphism or isogeny classes of maximal
K-tori. This leads to the necessary and sufficient conditions for two
Zariski-dense S-arithmetic subgroups of G and G' to be weakly commensurable
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
Modular Lie algebras and the Gelfand-Kirillov conjecture
Let g be a finite dimensional simple Lie algebra over an algebraically closed
field of characteristic zero. We show that if the Gelfand-Kirillov conjecture
holds for g, then g has type A_n, C_n or G_2.Comment: 20 page
Few smooth d-polytopes with n lattice points
We prove that, for fixed n there exist only finitely many embeddings of
Q-factorial toric varieties X into P^n that are induced by a complete linear
system. The proof is based on a combinatorial result that for fixed nonnegative
integers d and n, there are only finitely many smooth d-polytopes with n
lattice points. We also enumerate all smooth 3-polytopes with at most 12
lattice points. In fact, it is sufficient to bound the singularities and the
number of lattice points on edges to prove finiteness.Comment: 20+2 pages; major revision: new author, new structure, new result
Fano Varieties in Mori Fibre Spaces
We show that being a general fibre of a Mori fibre space is a rather
restrictive condition for a Fano variety. More specifically, we obtain two
criteria (one sufficient and one necessary) for a Q-factorial Fano variety with
terminal singularities to be realised as a fibre of a Mori fibre space, which
turn into a characterisation in the rigid case. We apply our criteria to figure
out this property up to dimension three and on rational homogeneous spaces. The
smooth toric case is studied and an interesting connection with K-semistability
is also investigated
Neutrino emission from dense matter, and neutron star thermal evolution
A brief review is given of neutrino emission processes in dense matter, with particular emphasis on recent developments. These include direct Urca processes for nucleons and hyperons, which can give rise to rapid energy loss from the stellar core without exotic matter, and the effect of band structure on neutrino bremsstrahlung from electrons in the crust, which results in much lower energy losses by this process than had previously been estimated