Minimal Affine Research Articles

Colour valued scattering matrices

We describe a general construction principle which allows to add colour values to a coupling constant dependent scattering matrix. As a concrete realization of this mechanism we provide a new type of S-matrix which generalizes the one of affine Toda field theory, being related to a pair of Lie algebras. A characteristic feature of this S-matrix is that in general it violates parity invariance. For particular choices of the two Lie algebras involved this scattering matrix coincides with the one related to the scaling models described by the minimal affine Toda S-matrices and for other choices with the one of the Homogeneous sine-Gordon models with vanishing resonance parameters. We carry out the thermodynamic Bethe ansatz and identify the corresponding ultraviolet effective central charges.

On fixed-length confidence intervals for a bounded normal mean

Consider the problem of estimating the mean of a single normal random variable when the mean is known to be bounded. We establish the minimax affine estimator under zero-one loss and discuss minimal fixed-length affine confidence intervals. Moreover, the minimal length of arbitrary fixed-size confidence intervals is described and the maximal loss of efficiency caused by the restriction to affine estimators is determined.

Minimal Affinizations of Representations of Quantum Groups: The Simply Laced Case

If Uq(g) is a finite-dimensional complex simple Lie algebra, an affinization of a finite-dimensional irreducible representationV of Uq(g) is a finite-dimensional irreducible representation\(\hat V\) of Uq(ĝ) which containsV with multiplicity one, and is such that all other Uq(g)-types in\(\hat V\) have highest weights strictly smaller than that ofV. There is a natural partial ordering\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \prec } \) on the set of affinizations, defined by Chari. In this Letter, we describe the minimal affinizations, with respect to\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \prec } \), when g is not simply-laced.

Minimal affinizations of representations of quantum groups: the irregular case

Let\(\mathfrak{g}\) be a finite-dimensional complex simple Lie algebra and Uq(\(\mathfrak{g}\)) the associated quantum group (q is a nonzero complex number which we assume is transcendental). IfV is a finitedimensional irreducible representation of Uq(\(\mathfrak{g}\)), an affinization ofV is an irreducible representationVV of the quantum affine algebra Uq(\(\hat {\mathfrak{g}}\)) which containsV with multiplicity one and is such that all other irreducible Uq(\(\mathfrak{g}\))-components ofV have highest weight strictly smaller than the highest weight λ ofV. There is a natural partial order on the set of Uq(\(\mathfrak{g}\)) classes of affinizations, and we look for the minimal one(s). In earlier papers, we showed that (i) if\(\mathfrak{g}\) is of typeA, B, C, F orG, the minimal affinization is unique up to Uq(\(\mathfrak{g}\))-isomorphism; (ii) if\(\mathfrak{g}\) is of typeD orE and λ is not orthogonal to the triple node of the Dynkin diagram of\(\mathfrak{g}\), there are either one or three minimal affinizations (depending on λ). In this paper, we show, in contrast to the regular case, that if Uq(\(\mathfrak{g}\)) is of typeD4 and λ is orthogonal to the triple node, the number of minimal affinizations has no upper bound independent of λ.

Minimal Affinizations of Representations of Quantum Groups: the Rank 2 Case

If U_q (\mathfrak g) is a finite-dimensional complex simple Lie algebra, an affinization of a finite-dimensional irreducible representation V of U_q (\mathfrak g) is a finite-dimensional irreducible representation \hat V of U_q (\mathfrak g) which contains \hat V with multiplicity one, and is such that all other U_q (\mathfrak g) -types in \hat V have highest weights strictly smaller than that of V . We define a natural partial ordering ≼ on the set of affinizations of V . If \mathfrak g is of rank 2 , we show that there is a unique minimal element with respect to this order and give its U_q (\mathfrak g) -module structure when \mathfrak g is of type A_2 or C_2 .

Minimal affinizations of representations of quantum groups: The nonsimply-laced case

If U q (g) is a finite-dimensional complex simple Lie algebra, an affinization of a finite-dimensional irreducible representationV of U q (g) is a finite-dimensional irreducible representation\(\hat V\) of U q (ĝ) which containsV with multiplicity one, and is such that all other U q (g)-types in\(\hat V\) have highest weights strictly smaller than that ofV. There is a natural partial ordering\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \prec } \) on the set of affinizations, defined by Chari. In this Letter, we describe the minimal affinizations, with respect to\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \prec } \), when g is not simply-laced.

Equidecomposable and weakly neighborly polytopes

A polytope is equidecomposable if all its triangulations have the same face numbers. For an equidecomposable polytope all minimal affine dependencies have an equal number of positive and negative coefficients. A subclass consists of the weakly neighborly polytopes, those for which every set of vertices is contained in a face of at most twice the dimension as the set. Theh-vector of every triangulation of a weakly neighborly polytope equals theh-vector of the polytope itself. Combinatorial properties of this class of polytopes are studied. Gale diagrams of weakly neighborly polytopes with few vertices are characterized in the spirit of the known Gale diagram characterization of Lawrence polytopes, a special class of weakly neighborly polytopes.

Minimal Affine Boundaries of Convex Sets

Minimal Affine Boundaries of Convex Sets

K-theoretic invariants for C∗-algebras associated to transformations and induced flows

We discuss a technique of studying the K-theory of a unital C ∗-algebra associated to a homomorphism on a compact metric space ( Y Z ) by examining the non-unital C ∗-algebra associated to the induced topological flow (Ind Z R ( Y), R ). The Thom isomorphism of Connes and the Schwartzman asymptotic cycle are used to calculate the range of the trace corresponding to an invariant measure on the K 0 group of C ∗( X, R ) for a continuous flow on a compact metric space ( X, R ). Under certain conditions projections in C ∗( X, R ) with trace r corresponding to cross sections to the flow can be constructed for every positive real number r in this range, again by combining techniques of Connes and Schwartzman. Applications to the calculation of the tracial range of K 0( C ∗( X, R )) are discussed. In particular, this invariant is calculated for minimal affine actions of Z on n-tori which have quasi-discrete spectrum, and for minimal actions of Z on compact abelian groups with topologically discrete spectrum. In both cases this tracial invariant is shown to be the preimage of the eigenvalues for ( Y, Z ) under the natural projection ø: R → R / Z = S 1.

Compact abelian group extensions of discrete dynamical systems

This paper introduces the notion of a free G extension of a dynamical system where G is a compact abelian group. The concept is closely allied to that of generalised discrete spectrum (which includes Abramov's quasi-discrete spectrum as a special case). We give necessary and sufficient conditions for a G extension of a minimal (uniquely ergodic) dynamical system to be minimal (uniquely ergodic) and show that in a certain sense a general G extension lifts these properties. Stable G-extensions always lift these properties if the underlying space is connected. This fact is then used to characterise all uniquely ergodic and minimal affine transformations of a certain three dimensional nilmanifold. The rest of the paper is devoted to the exhibition of group invariants for systems with generalised discrete spectrum. In particular it is shown that such systems always have a compact abelian group as underlying space. A lemma which facilitates this result gives necessary and sufficient conditions for a connected G-extension of a compact abelian group to be a compact abelian group.

Topological conjugacy of affine transformations of tori

0. Introduction. Let Kn denote the n-dimensional torus. Let T be an affine transformation of KI onto Kn and S an affine transformation of Km onto Km for which there exists a continuous map h of Kn onto Km satisfying hT=Sh. R. L. Adler and R. Palais have proved that, in the special cases when m = n and T and S are ergodic automorphisms of Kn, all homeomorphism g of Kn, satisfying gT= Sg, are affine transformations [1]. The main purpose of this paper is to obtain necessary and sufficient conditions for the existence of a nonaffine continuous map g of Kn onto Km such that gT= Sg. (Theorem 2, ?4). The conditions obtained are unchanged under the requirements that m =n and T, S, h and g be invertible transformations of Kn. We obtain the above-mentioned result of Adler and Palais as a corollary of Theorem 2. We also prove that two automorphisms T and S of Kn must be ergodic if all homeomorphisms g of Kn such that gT= Sg are affine transformations. (We require, of course, that there exist homeomorphisms g such that gT= Sg.) Another corollary of Theorem 2 concerns affine transformations with quasi-discrete spectrum. A. H. M. Hoare and W. Parry have shown that every minimal affine transformation of a compact, connected, metric, abelian group X has quasidiscrete spectrum, and that if T and S are minimal affine transformations of X then all homeomorphisms g of X, such that gT= Sg, are affine [5], [6]. Our corollary is restricted to toroidal groups but extends Hoare and Parry's result in this case. Theorem 2 is also used to obtain a necessary and sufficient condition for a strong-mixing affine transformation of Kn to have a continuous pth root (Theorem 3, ?5). In Theorem 1 a necessary and sufficient condition for an affine transformation of a compact, connected, metric, abelian group to be ergodic is given. I should like to express my gratitude to Dr. W. Parry for supervising this work and also to Dr. D. E. Cohen and Dr. A. H. M. Hoare for supplying useful examples. My thanks are also due to the referee for some helpful suggestions.