Basic coset geometries

In earlier work we gave a characterisation of pregeometries which are ‘basic’ (that is, admit no ‘non-degenerate’ quotients) relative to two different kinds of quotient operation, namely taking imprimitive quotients and normal quotients. Each basic geometry was shown to involve a faithful group action, which is primitive or quasiprimitive, respectively, on the set of elements of each type. For each O’Nan-Scott type of primitive group, we construct a new infinite family of geometries, which are thick and of unbounded rank, and which admit a flag-transitive automorphism group acting faithfully on the set of elements of each type as a primitive group of the given O’Nan-Scott type.     

Weights and covers of chamber systems

The notion of weight algebra for a group geometry is developed for flag-transitive subgroups and their quotients, as a technique for establishing the existence of weight vectors in modular representations. This extends the applicability of earlier results, in particular results for the full group of a spherical building. The method adapts for at least some of the sporadic geometries provided by finite quotients of infinite affine buildings. Dedicated to Professor Jacques Tits on his 60th birthday Partially supported by NSF grant DMS 88-01679 and NSA grant MDA 904-87-H-2006.     

Extending Ring Derivations to Right and Symmetric Rings and Modules of Quotients

We define and study the symmetric version of differential torsion theories. We prove that the symmetric versions of some of the existing results on derivations on right modules of quotients hold for derivations on symmetric modules of quotients. In particular, we prove that the symmetric Lambek, Goldie, and perfect torsion theories are differential.

We also study conditions under which a derivation on a right or symmetric module of quotients extends to a right or symmetric module of quotients with respect to a larger torsion theory. Using these results, we study extensions of ring derivations to maximal, total, and perfect right and symmetric rings of quotients.     

Cartan Geometries and Dynamics

We show how some basic dynamical ideas can be brought to bear on the study of Cartan geometries. In our main results, we give conditions for certain types of Cartan geometries to have constant curvature. We also consider ergodic actions of `higher-rank' semisimple groups on bundles supporting (not necessarily invariant) Cartan connections and show that these are `standard locally homogeneous' actions provided that some noncompact 1-parameter subgroup `preserves' a Cartan geometry.     

Characterizations of the convex geometries arising from the double shellings of posets

We investigate the class of double-shelling convex geometries. A double-shelling convex geometry is the collection of sets represented as the intersection of an ideal and a filter of a poset. The size of the stem of any rooted circuit of a double-shelling convex geometry is 2. We characterize the double-shelling convex geometries by the conditions that the rooted circuits should fulfill. Moreover we also characterize the same class in terms of trace-minimal forbidden minors.     

A theorem on tits geometries of type Cn

In this paper we extend a result of [2] to cover a more general situation. In [2] it was shown that a finite C_n geometry (n 4) in which all lines are thick and all C₃ residues are either buildings or flat has to be a building. Here we observe that the finiteness assumption of [2] was unnecessary in order to achieve the major part of the result. If we drop the finiteness assumption we can still prove that such a geometry is either a quotient of a building or flat. Flat C_n geometries for n 4 are seen to be degenerate in a certain sense. In the finite case with thick lines such degenerate geometries are easily shown not to exist, while finite buildings of type C_n with thick lines do not admit non-trivial quotients (Brouwer and Cohen, [1]). Thus the result of [2] follows as an immediate corollary of this more general case. The result does not hold when we drop the assumption that all lines are thick. In Section 3 we produce some examples of geometries of this type.     

A geometric Schur–Weyl duality for quotients of affine Hecke algebras

After establishing a geometric Schur–Weyl duality in a general setting, we recall this duality in type A in the finite and affine case. We extend the duality in the affine case to positive parts of the affine algebras. The positive parts have nice ideals coming from geometry, allowing duality for quotients. Some of the quotients of the positive affine Hecke algebra are then identified to some cyclotomic Hecke algebras and the geometric setting allows the construction of canonical bases.     

Interpolatory wavelets for manifold-valued data

Geometric wavelet-like transforms for univariate and multivariate manifold-valued data can be constructed by means of nonlinear stationary subdivision rules which are intrinsic to the geometry under consideration. We show that in an appropriate vector bundle setting for a general class of interpolatory wavelet transforms, which applies to Riemannian geometry, Lie groups and other geometries, Hölder smoothness of functions is characterized by decay rates of their wavelet coefficients.     

Symmetric parabolic contact geometries and symmetric spaces

We discuss parabolic contact geometries carrying a smooth system of symmetries. We show that there is a symmetric space such that the parabolic geometry $ \left( {\mathcal{G}\to M,\omega } \right) $ is a fibre bundle over this symmetric space if and only if the base manifold M is a homogeneous reexion space. We investigate the conditions under which there is an invariant geometric structure on this symmetric space induced by the parabolic contact geometry.     

Categorical Aspects in Projective Geometry

After introducing morphisms between projective geometries, some categorical questions are examined. It is shown that there are three kinds of embeddings and two kinds of quotients. Furthermore the morphisms decompose in a canonical way into four factors.     

On Af.Af* geometries

This paper is intended to be a first step towards the classification of finite flag-transitive geometries of rank 3 with affine planes and dual affine point-residues. We describe those of diameter 1. In the case of diameter > 1, we describe minimal quotients, assuming that the number of lines through two points is large enough.     

Centers,cocenters and simple quantum groups

We define the notion of a (linearly reductive) center for a linearly reductive quantum group, and show that the quotient of a such a quantum group by its center is simple whenever its fusion semiring is free in the sense of Banica and Vergnioux. We also prove that the same is true of free products of quantum groups under very mild non-degeneracy conditions. Several natural families of compact quantum groups, some with non-commutative fusion semirings and hence very “far from classical”, are thus seen to be simple. Examples include quotients of free unitary groups by their centers, recovering previous work, as well as quotients of quantum reflection groups by their centers.     

Proving the triviality of rational points on Atkin–Lehner quotients of Shimura curves

In this paper we give a method for studying global rational points on certain quotients of Shimura curves by Atkin–Lehner involutions. We obtain explicit conditions on such quotients for rational points to be “trivial” (coming from CM points only) and exhibit an explicit infinite family of such quotients satisfying these conditions.     

On Flat Flag-Transitive c.c *-Geometries

We study flat flag-transitive c.c ^*-geometries. We prove that, apart from one exception related to Sym(6), all these geometries are gluings in the meaning of [6]. They are obtained by gluing two copies of an affine space over GF(2). There are several ways of gluing two copies of the n-dimensional affine space over GF(2). In one way, which deserves to be called the canonical one, we get a geometry with automorphism group G = 2²ⁿ · L _n(2) and covered by the truncated Coxeter complex of type D ₂ ⁿ . The non-canonical ways give us geometries with smaller automorphism group (G ≤ 2²ⁿ · (2 ^n?1)n) and which seldom (never ?) can be obtained as quotients of truncated Coxeter complexes.     

Some results on cosymplectic manifolds

We obtain a generalization of the Kodaira-Morrow stability theorem for cosymplectic structures. We investigate cosymplectic geometry on Lie groups and on their compact quotients by uniform discrete subgroups. In this way we show that a compact solvmanifold admits a cosymplectic structure if and only if it is a finite quotient of a torus.     

Projective representations II. generalized chain geometries

In this paper, projective representations of generalized chain geometries are investigated, using the concepts and results of [5]. In particular, we study under which conditions such a projective representation maps the chains of a generalized chain geometry Σ (F, R) to reguli; this mainly depends on how the field F is embedded in the ring R. Moreover, we determine all bijective morphisms of a certain class of generalized chain geometries with the help of projective representations. Dedicated to Walter Benz on the occasion of his 70th birthday. The first author was supported by a Lise Meitner Research Fellowship of the Austrian Science Fund (FWF), projects M529-MAT, M574-MAT..     

Small Amplitude Theory of Richtmyer—Meshkov Instability in Cylindrical and Spherical Geometries

In this paper we formulate the linear theory for compressible fluids in spherical geometry. We derive the first-order equations in the smooth regions, boundary conditions at the shock fronts and the contact interface by linearizing the Euler equations and Rankine—Hugoniot conditions. This formulation can be used for the computation of the linear theory in spherical and cylindrical geometries.