7,001 research outputs found
Algebraic group actions on noncommutative spectra
Let G be an affine algebraic group and let R be an associative algebra with a
rational action of G by algebra automorphisms. We study the induced G-action on
the spectrum Spec R of all prime ideals of R, viewed as a topological space
with the Jacobson-Zariski topology, and on the subspace consisting of all
rational ideals of R. Here, a prime ideal P of R is said to be rational if the
extended centroid of R/P is equal to the base field. The main themes of the
article are local closedness of G-orbits in Spec R and the so-called
G-stratification of Spec R. This stratification plays a central role in the
recent investigation of algebraic quantum groups, in particular in the work of
Goodearl and Letzter. We describe the G-strata in terms of certain commutative
spectra. Our principal results are based on prior work of Moeglin & Rentschler
and Vonessen. We generalize the theory arbitrary associative algebras while
also simplifying some of the earlier proofs.Comment: 21 pages; minor revisions; to appear in Transformation Group
On the Cohen-Macaulay property of multiplicative invariants
We investigate the Cohen-Macaulay property for rings of invariants under
multiplicative actions of a finite group . By definition, these are
-actions on Laurent polynomial algebras that stabilize the multiplicative
group consisting of all monomials in the variables. For the most part, we
concentrate on the case where the base ring is the ring of rational integers.
Our main result states that if acts non-trivially and the invariant algebra
is Cohen-Macaulay then the abelianized isotropy groups of all
monomials m are generated by bireflections and at least one is
non-trivial. As an application, we prove the multiplicative version of Kemper's
3-copies conjecture.Comment: 16 pages, LaTeX; some new results and examples added; expanded
introduction with additional reference
On Euler classes of abelian-by-finite groups
Let be a finitely generated abelian-by-finite group and a field of
characteristic . The Euler class of over is the class
of the trivial -module in the Grothendieck group . We show that
has finite order if and only if every -regular element of has
infinite centralizer in . We also give a lower bound for the order of the
Euler class in terms of suitable finite subgroups of . This lower bound is
derived from a more general result on finite-dimensional representations of
smash products of Hopf algebras.Comment: 12 pages, 2 figures, AMSLaTe
Some applications of Frobenius algebras to Hopf algebras
This expository article presents a unified ring theoretic approach, based on
the theory of Frobenius algebras, to a variety of results on Hopf algebras.
These include a theorem of S. Zhu on the degrees of irreducible
representations, the so-called class equation, the determination of the
semisimplicity locus of the Grothendieck ring, the spectrum of the adjoint
class and a non-vanishing result for the adjoint character.Comment: 22 page
On the stratification of noncommutative prime spectra
We study rational actions of an algebraic torus G by automorphisms on an
associative algebra R. The G-action on R induces a stratification of the prime
spectrum of R which was introduced by Goodearl and Letzter. For a noetherian
algebra R, Goodearl and Letzter showed that the strata of the spectrum of R are
isomorphic to the spectra of certain commutative Laurent polynomial algebras.
The purpose of this note is to give a new proof of this result which works for
arbitrary algebras R
On the degrees of irreducible representations of Hopf algebras
Let H denote a semisimple Hopf algebra over an algebraically closed field k
of characteristic 0. We show that the degree of any irreducible representation
of H whose character belongs to the center of H^* must divide the dimension of
H .Comment: AMS-LaTeX, 3 page
K_0 of invariant rings and nonabelian H^1
We give a description of the kernel of the induction map K_0(R)->K_0(S),
where S is a commutative ring and R is the ring of invariants of the action of
a finite group G on S. The description is in terms of H^1(G,GL(S))
Bochner-Kaehler metrics and connections of Ricci type
We apply the results from the article Cahen, Schwachh\"ofer: Special
symplectic connections, to the case of Bochner-Kaehler metrics. We obtain a
(local) classification of these based on the orbit types of the adjoint action
in . The relation between Sasaki and Bochner-Kaehler metrics in cone
and transveral metrics constructions is discussed. The connection of the
special symplectic and Weyl connections is outlined. The duality between the
Ricci-type and Bochner-Kaehler metrics is shown
Bipolar Electrodiffusion model for Electroconvection in Nematics
The common description of the electrical behavior of a nematic liquid crystal
as an anisotropic dielectric medium with (weak) ohmic conductivity is extended
to an electrodiffusion model with two active ionic species. Under appropriate,
but rather general conditions the additional effects can lead to a distinctive
change of the threshold behavior of the electrohydrodynamic instability, namely
to travelling patterns instead of static ones. This may explain the
experimentally observed phenomena.Comment: 15 pages, 3 figures (postscript
On certain lattices associated with generic division algebras
Let S_n denote the symmetric group on n letters. We consider the S_n-root
lattice A_{n-1} = {(z1,...,zn) in Z^n | z1+...+zn = 0}, where S_n acts on Z^n
by permuting the coordinates, and its tensor, symmetric, and exterior squares.
For odd values of n, we show that the tensor square is equivalent, in the sense
of Colliot-Thelene and Sansuc, to the exterior square. Consequently, the
rationality problem for generic division algebras, for odd values of n, amounts
to proving stable rationality of the multiplicative S_n-invariant field of the
exterior square of A_{n-1}. Furthermore, confirming a conjecture of Le Bruyn,
we show that n=2 and n=3 are the only cases where the tensor square of A_{n-1}
is equivalent to a permutation S_n-lattice. In the course of the proof of this
result, we construct subgroups H of S_n, for all n that are not prime, so that
the algebra of multiplicative H-invariants of A_{n-1} has a non-trivial Picard
group.Comment: 19 pages, AMS-LaTeX with XyPi
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