Three-Dimensional Classical Groups Acting on Polytopes

Let V be a three-dimensional vector space over a finite field. We show that any irreducible subgroup of GL(V) that arises as the automorphism group of an abstract regular polytope preserves a nondegenerate symmetric bilinear form on V. In particular, the only classical groups on V that arise as automorphisms of such polytopes are the orthogonal groups.     

Lie algebras of infinitesimal norm isometries

Given a norm on a finite dimensional vector space V, we may consider the group of all linear automorphisms which preserve it. The Lie algebra of this group is a Lie subalgebra of the endomorphism algebra of V having two properties: (1) it is the Lie algebra of a compact subgroup, and (2) it is “saturated” in a sence made precise below. We show that any Lie subalgebra satisfying these conditions is the Lie algebra of the group of linear automorphisms preserving some norm. There is an appendix on elementary Lie group theory.     

Lie group structures on symmetry groups of principal bundles

In this paper we describe how one can obtain Lie group structures on the group of (vertical) bundle automorphisms for a locally convex principal bundle P over the compact manifold M. This is done by first considering Lie group structures on the group of vertical bundle automorphisms Gau(P). Then the full automorphism group Aut(P) is considered as an extension of the open subgroup DiffP(M) of diffeomorphisms of M preserving the equivalence class of P under pull-backs, by the gauge group Gau(P). We derive explicit conditions for the extensions of these Lie group structures, show the smoothness of some natural actions and relate our results to affine Kac-Moody algebras and groups.     

The Rohlin property for shifts of finite type

We show that an automorphism of a unital AF C^*-algebra with a certain approximate Rohlin property has the Rohlin property. This generalizes a result of Kishimoto. Using this we show that the shift automorphism on the bilateral C^*-algebra associated with an aperiodic irreducible shift of finite type has the Rohlin property.     

Classification of 9-dimensional trilinear alternating forms over GF(2)

Let V be a finite-dimensional vector space over a finite field and let f be a trilinear alternating form over V. For such forms, we introduce two new invariants. Together with a generalized radical polynomial used for classification of forms in dimension 8 over GF(2), they are sufficient to distinguish between all trilinear alternating forms in dimension 9 over GF(2). To prove the completeness of the list of forms, we computed their groups of automorphisms. There are 31 degenerate and 317 nondegenerate forms. We point out some forms with either small or large automorphism group.     

The Lipschitzianity of convex vector and set-valued functions

It is well known that every scalar convex function is locally Lipschitz on the interior of its domain in finite dimensional spaces. The aim of this paper is to extend this result for both vector functions and set-valued mappings acting between infinite dimensional spaces with an order generated by a proper convex cone C. Under the additional assumption that the ordering cone C is normal, we prove that a locally C-bounded C-convex vector function is Lipschitz on the interior of its domain by two different ways. Moreover, we derive necessary conditions for Pareto minimal points of vector-valued optimization problems where the objective function is C-convex and C-bounded. Corresponding results are derived for set-valued optimization problems.     

Automorphismen Von Cayleyalgebren Über Algebraischen Erweiterungen Endlicher KÖrper Ungerader Charakteristik

Let C denote the (split) Cayley algebra over a finite field K of odd characteristic. Given any automorphism σ of C, which is not expressible as the product of two involutory automorphisms, we show that the minimal polynomial of σ is (x ? l)(x² + x + 1)³]. This result remains true, if K is replaced by an infinite algebraic extension K′ of K. Furthermore the automorphism group of C over K′ is bireflectional iff every polynomial of degree 3 in K′[x] is reducible. This corrects and extends the results achieved by Huberta Lausch in [2].     

Classification of non-degenerate projective varieties with non-zero prolongation and application to target rigidity

The prolongation \(\mathfrak{g}^{(k)}\) of a linear Lie algebra \(\mathfrak{g}\subset \mathfrak{gl}(V)\) plays an important role in the study of symmetries of G-structures. Cartan and Kobayashi-Nagano have given a complete classification of irreducible linear Lie algebras \(\mathfrak{g}\subset \mathfrak{gl}(V)\) with non-zero prolongations.If \(\mathfrak{g}\) is the Lie algebra \(\mathfrak{aut}(\hat{S})\) of infinitesimal linear automorphisms of a projective variety S??V, its prolongation \(\mathfrak{g}^{(k)}\) is related to the symmetries of cone structures, an important example of which is the variety of minimal rational tangents in the study of uniruled projective manifolds. From this perspective, understanding the prolongation \(\mathfrak{aut}(\hat{S})^{(k)}\) is useful in questions related to the automorphism groups of uniruled projective manifolds. Our main result is a complete classification of irreducible non-degenerate nonsingular variety S??V with \(\mathfrak{aut}(\hat {S})^{(k)}\neq0\), which can be viewed as a generalization of the result of Cartan and Kobayashi-Nagano. As an application, we show that when S is linearly normal and Sec?(S)≠?V, the blow-up Bl _S (?V) has the target rigidity property, i.e., any deformation of a surjective morphism f:Y→Bl _S (?V) comes from the automorphisms of Bl _S (?V).     

An interpolation theorem for holomorphic automorphisms ofC n

We construct automorphisms of C ⁿ which map certain discrete sequences one onto another with prescribed finite jet at each point, thus solving a general Mittag-Leffler interpolation problem for automorphisms. Under certain circumstances, this can be done while also approximating a given automorphism on a compact set.     

Enveloppes polynômiales d'ensembles compacts invariants

We consider the following problem: let V^? be a finite dimensional vector space, and U be a compact group of ?‐linear automorphisms of V^?. The polynomial envelope of a compact set Q ? V^? is defined as where ??(V^?) denotes the space of holomorphic polynomial functions on V^?. The problem is to determine the polynomial envelope of a compact set which is U‐invariant. We solve the problem when U is the isotropy subgroup at the origin of the automorphism group of a bounded symmetric domain of tube type. The case of a domain of type II has been solved by C. Sacré [1992], and, for a domain of type IV, it has been solved by L. Bou Attour [1993]. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Common homoclinic points of commuting toral automorphisms

The points homoclinic to 0 under a hyperbolic toral automorphism form the intersection of the stable and unstable manifolds of 0. This is a subgroup isomorphic to the fundamental group of the torus. Suppose that two hyperbolic toral automorphisms commute so that they determine a ℤ²-action, which we assume is irreducible. We show, by an algebraic investigation of their eigenspaces, that they either have exactly the same homoclinic points or have no homoclinic point in common except 0 itself. We prove the corresponding result for a compact connected abelian group, and compare the two proofs. The second author would like to thank the Austrian Academy of Sciences and the Royal Society for partial support while this work was done.     

Twisted modules over vertex algebras on algebraic curves

We extend the geometric approach to vertex algebras developed by the first author to twisted modules, allowing us to treat orbifold models in conformal field theory. Let V be a vertex algebra, H a finite group of automorphisms of V, and C an algebraic curve such that H⊂Aut(C). We show that a suitable collection of twisted V-modules gives rise to a section of a certain sheaf on the quotient X=C/H. We introduce the notion of conformal blocks for twisted modules, and analyze them in the case of the Heisenberg and affine Kac-Moody vertex algebras. We also give a chiral algebra interpretation of twisted modules.     

Lattice path combinatorics and asymptotics of multiplicities of weights in tensor powers

We give asymptotic formulas for the multiplicities of weights and irreducible summands in high-tensor powers V_λ^⊗N of an irreducible representation V_λ of a compact connected Lie group G. The weights are allowed to depend on N, and we obtain several regimes of pointwise asymptotics, ranging from a central limit region to a large deviations region. We use a complex steepest descent method that applies to general asymptotic counting problems for lattice paths with steps in a convex polytope.     

Nil-automorphisms of groups with residual properties

Let G be any group and x an automorphism of G. The automorphism x is said to be nil if, for every g ∈ G, there exists n = n(g) such that [g, _n x] = 1. If n can be chosen independently of g, we say that x is n-unipotent. A nil (resp. unipotent) automorphism x could also be seen as a left Engel element (resp. left n-Engel element) in the group G〈x〉. When G is a finite dimensional vector space, groups of unipotent linear automorphisms turn out to be nilpotent, so that one might ask to what extent this result can be extended to a more general setting. In this paper we study finitely generated groups of nil or unipotent automorphisms of groups with residual properties (e.g. locally graded groups, residually finite groups, profinite groups), proving that such groups are nilpotent.     

The Rohlin property for automorphisms of the Jiang-Su algebra

For projectionless C^∗-algebras absorbing the Jiang-Su algebra tensorially, we study a kind of the Rohlin property for automorphisms. We show that the crossed products obtained by automorphisms with this Rohlin property also absorb the Jiang-Su algebra tensorially under a mild technical condition on the C^∗-algebras. In particular, for the Jiang-Su algebra we show the uniqueness up to outer conjugacy of the automorphism with this Rohlin property.     

On the Stability of the Motzkin Representation of Closed Convex Sets

A set is called Motzkin decomposable when it can be expressed as the Minkowski sum of a compact convex set with a closed convex cone. This paper analyzes the continuity properties of the set-valued mapping associating to each couple $\left( C,D\right) $ formed by a compact convex set C and a closed convex cone D its Minkowski sum C?+?D. The continuity properties of other related mappings are also analyzed.     

Symbolic representations of nonexpansive group automorphisms

Ifα is an irreducible nonexpansive ergodic automorphism of a compact abelian groupX (such as an irreducible nonhyperbolic ergodic toral automorphism), thenα has no finite or infinite state Markov partitions, and there are no nontrivial continuous embeddings of Markov shifts inX. In spite of this we are able to construct a symbolic spaceV and a class of shift-invariant probability measures onV each of which corresponds to anα-invariant probability measure onX. Moreover, everyα-invariant probability measure onX arises essentially in this way. The last part of the paper deals with the connection between the two-sided beta-shiftV _β arising from a Salem numberβ and the nonhyperbolic ergodic toral automorphismα arising from the companion matrix of the minimal polynomial ofβ, and establishes an entropy-preserving correspondence between a class of shift-invariant probability measures onV _β and certainα-invariant probability measures onX. This correspondence is much weaker than, but still quite closely modelled on, the connection between the two-sided beta-shifts defined by Pisot numbers and the corresponding hyperbolic ergodic toral automorphisms.     

Involutions of Type G<Subscript>2</Subscript> Over Fields of Characteristic Two

We continue a study of automorphisms of order 2 of algebraic groups. In particular we look at groups of type G₂ over fields k of characteristic two. Let C be an octonion algebra over k; then Aut(C) is a group of type G₂ over k. We characterize automorphisms of order 2 and their corresponding fixed point groups for Aut(C) by establishing a connection between the structure of certain four dimensional subalgebras of C and the elements in Aut(C) that induce inner automorphisms of order 2. These automorphisms relate to certain quadratic forms which, in turn, determine the Galois cohomology of the fixed point groups of the involutions. The characteristic two case is unique because of the existence of four dimensional totally singular subalgebras. Over finite fields we show how our results coincide with known results, and we establish a classification of automorphisms of order 2 over infinite fields of characteristic two.     

Noncompact simplexes in Banach spaces with the Radon-Nikodým property

It is well known that a compact convex subset C of a locally convex topological vector space is a simplex if and only if each point x of C admits a unique probability measure on the extreme points of C with barycenter x. An exact analog of this result is proved for a closed and bounded separable convex subset of a Banach space with the Radon-Nikodým Property, and a weaker analog is proved in the nonseparable case.