The Grothendieck conjecture for affine curves

We prove that the isomorphism class of an affine hyperbolic curve defined over a field finitely generated over Q is completely determined by its arithmetic fundamental group. We also prove a similar result for an affine curve defined over a finite field.     

Affine complete varieties are congruence distributive

An algebra is affine complete iff its polynomial operations are the same as all the operations over its universe that are compatible with all its congruences. A variety is affine complete iff all its algebras are. We prove that every affine complete variety is congruence distributive, and give a useful characterization of all arithmetical, affine complete varieties of countable type. We show that affine complete varieties with finite residual bound have enough injectives. We also construct an example of an affine complete variety without finite residual bound.? We prove several results concerning residually finite varieties whose finite algebras are congruence distributive, while leaving open the question whether every such variety must be congruence distributive. Received February 28, 1997; accepted in final form December 9, 1997.     

On the Automorphism Groups of Affine-Invariant Codes.

Using the classification of finite simple groups, we obtain a condition for a permutation group on a finite field GF(pm) to have the affine group AGL(1,pm) as a subgroup. Applying this result to the automorphism groups of non trivial affine-invariant codes, we prove that these automorphism groups are always subgroups of the general affine group AGL(m,p).     

Current algebras, highest weight categories and quivers

We study the category of graded finite-dimensional representations of the polynomial current algebra associated to a simple Lie algebra. We prove that the category has enough injectives and compute the graded character of the injective envelopes of the simple objects as well as extensions between simple objects. The simple objects in the category are parametrized by the affine weight lattice. We show that with respect to a suitable refinement of the standard ordering on the affine weight lattice the category is highest weight. We compute the Ext quiver of the algebra of endomorphisms of the injective cogenerator of the subcategory associated to an interval closed finite subset of the weight lattice. Finally, we prove that there is a large number of interesting quivers of finite, affine and star-shaped type, as well as tame quasi-hereditary algebras, that arise from our study.     

Piecewise affine functions and polyhedral sets ∗

In this paper we present a number of characterizations of piecewise affine and piecewise linear functions defined on finite dimesional normed vector spaces. In particular we prove that a real-valued function is piecewise affine [resp. piecewise linear] if both its epigraph and its hypograph are (nonconvex) polyhedral sets[resp..Polyhedral cones]. Also,We show that the collection of all piecewise affine[resp.piecewise linear] functions. Furthermore, we prove that a function is piecewise affine[resp.piecewise linear] if it can be represented as a difference of two convex [resp.,sublinear] polyhedral fucntions.     

Uniqueness of non-Archimedean entire functions sharing sets of values counting multiplicity

A set is called a unique range set for a certain class of functions if each inverse image of that set uniquely determines a function from the given class. We show that a finite set is a unique range set, counting multiplicity, for non-Archimedean entire functions if and only if there is no non-trivial affine transformation preserving the set. Our proof uses a theorem of Berkovich to extend, to non-Archimedean entire functions, an argument used by Boutabaa, Escassut, and Haddad to prove this result for polynomials

     

Affine semigroup rings are of finite F-representation type

It is known that normal affine semigroup rings are of finite F-representation type. In this paper, we prove that non-normal affine semigroup rings are also of finite F-representation type.     

On the Frobenius stable part of Witt vector cohomology

For a proper (not necessarily smooth) variety over a finite field with q elements, Berthelot?CBloch?CEsnault proved a trace formula which computes the number of rational points modulo q in terms of the Witt vector cohomology. We show the analogous formula for Witt vector cohomology of finite length. In addition, we prove a vanishing result for the compactly supported étale cohomology of a constant p-torsion sheaf on an affine Cohen?CMacaulay variety.     

Two-dimensional quotients of C<Superscript>n</Superscript> are isomorphic to C<Superscript>2</Superscript>/Γ

We will prove that any two-dimensional quotient of an affine space modulo a reductive algebraic group is isomorphic to a quotient of C² modulo a finite group. The proof uses some new results due to Koras and Russell on contractible surfaces with at most quotient singularities and also several results about reductive group actions on affine varieties.     

Closed Conjugacy Classes,Closed Orbits and Structure of Algebraic Groups

In this paper we prove that if an affine algebraic group (in characteristic zero) has all its conjugacy classes closed, then it is nilpotent. A classical result (called sometimes the Kostant-Rosenlicht Theorem) guarantees that if an affine algebraic group G is unipotent, then all its orbits on affine varieties are closed. We prove the converse of that theorem in arbitrary characteristics.     

Degenerate Affine Flag Varieties and Quiver Grassmannians

We study finite dimensional approximations to degenerate versions of affine flag varieties using quiver Grassmannians for cyclic quivers. We prove that they admit cellular decompositions parametrized by affine Dellac configurations, and that their irreducible components are normal Cohen-Macaulay varieties with rational singularities.

     

Twisted affine Lie superalgebra of type Q and quantization of its enveloping superalgebra

We introduce a new quantum group which is a quantization of the enveloping superalgebra of a twisted affine Lie superalgebra of type Q. We study generators and relations for superalgebras in the finite and twisted affine cases, and also universal central extensions. Afterwards, we apply the FRT formalism to a certain solution of the quantum Yang–Baxter equation to define that new quantum group and we study some of its properties. We construct a functor of Schur–Weyl type which connects it to affine Hecke–Clifford algebras and prove that it provides an equivalence between two categories of modules.     

Canonization theorems for finite affine and linear spaces

In this paper we prove a canonical (i.e. unrestricted) version of the Graham—Leeb—Rothschild partition theorem for finite affine and linear spaces [3]. We also mention some other kind of canonization results for finite affine and linear spaces.     

Regular sets for the affine and projective groups over the field of two elements

In this paper we show that collineation groups of affine and projective spaces over the field of two elements GF (2), except in low dimensions, have regular sets. As an application of this result, we prove that, apart from a finite number of exceptions, any collineation group of affine and projective spaces over GF (2), is geometric. In the exceptional dimensions, all primitive groups are examined.Lavoro eseguito nell'ambito dei finanziamenti del M.P.I. Italia (40%).     

Affine Poisson structures

We establish a one-to-one correspondence between the set of all equivalence classes of affine Poisson structures (defined on the dual of a finite dimensional Lie algebra) and the set of all equivalence classes of central extensions of the Lie algebra by ℝ. We characterize all the affine Poisson structures defined on the duals of some lower dimensional Lie algebras. It is shown that under a certain condition every Poisson structure locally looks like an affine Poisson structure. As an application, we show the role played by affine Poisson structures in mechanics. Finally, we prove some involution theorems.     

Block-transitive designs in affine spaces

This paper deals with block-transitive t-(v, k, λ) designs in affine spaces for large t, with a focus on the important index λ = 1 case. We prove that there are no non-trivial 5-(v, k, 1) designs admitting a block-transitive group of automorphisms that is of affine type. Moreover, we show that the corresponding non-existence result holds for 4-(v, k, 1) designs, except possibly when the group is 1-D affine. Our approach involves a consideration of the finite 2-homogeneous affine permutation groups.     

Noetherian PI Hopf algebras are Gorenstein

We prove that every noetherian affine PI Hopf algebra has finite injective dimension, which answers a question of Brown (1998).

     

Unital affine semigroups

Affine semigroups are convex sets on which there exists an associative binary operation which is affine separately in either variable. They were introduced by Cohen and Collins in 1959. We look at examples of affine semigroups which are of interest to matrix and operator theory and we prove some new results on the extreme points and the absorbing elements of certain types of affine semigroups. Most notably we improve a result of Wendel that every invertible element in a compact affine semigroup is extreme by extending this result to linearly bounded affine semigroups.     

Algebraicity of analytic maps to a hyperbolic variety

Let X be a complex algebraic variety. We say that X is Borel hyperbolic if, for every finite type reduced scheme S over the complex numbers, every holomorphic map from S to X is algebraic. We use a transcendental specialization technique to prove that X is Borel hyperbolic if and only if, for every smooth affine complex algebraic curve C, every holomorphic map from C to X is algebraic. We use the latter result to prove that Borel hyperbolicity shares many common features with other notions of hyperbolicity such as Kobayashi hyperbolicity.