Borchers’ Commutation Relations for Sectors with Braid Group Statistics in Low Dimensions

Borchers has shown that in a translation covariant vacuum representation of a theory of local observables with positive energy the following holds: The (Tomita) modular objects associated with the observable algebra of a fixed wedge region give rise to a representation of the subgroup of the Poincaré group generated by the boosts and the reflection associated to the wedge, and the translations. We prove here that Borchers’ theorem also holds in charged sectors with (possibly non-Abelian) braid group statistics in low space-time dimensions. Our result is a crucial step towards the Bisognano–Wichmann theorem for Plektons in d = 3, namely that the mentioned modular objects generate a representation of the proper Poincaré group, including a CPT operator. Our main assumptions are Haag duality of the observable algebra, and translation covariance with positive energy as well as finite statistics of the sector under consideration. Supported by FAPEMIG. Submitted: June 11, 2008., Accepted: August 4, 2008.     

Banach space representations and Iwasawa theory

We develop a duality theory between the continuous representations of a compactp-adic Lie groupG in Banach spaces over a givenp-adic fieldK and certain compact modules over the completed group ringo _K[[G]]. We then introduce a “finiteness” condition for Banach space representations called admissibility. It will be shown that under this duality admissibility corresponds to finite generation over the ringK[[G]]: =K ⊗o _K[[G]]. Since this latter ring is noetherian it follows that the admissible representations ofG form an abelian category. We conclude by analyzing the irreducibility properties of the continuous principal series of the groupG: = GL₂(ℤ _p ).     

Cohomology of tori over p-adic curves

 We establish a duality in the cohomology of arbitrary tori over smooth but not necessarily projective curves over a p-adic field. This generalises Lichtenbaum–Tate duality between the Picard group and the Brauer group of a smooth projective curve. Received: 28 January 2002 / Published online: 28 March 2003 Mathematics Subject Classification (2000): 14G20, 14F22, 14L15, 11S25     

Distributive Lattices with a Generalized Implication: Topological Duality

In this paper we introduce the notion of generalized implication for lattices, as a binary function ⇒ that maps every pair of elements of a lattice to an ideal. We prove that a bounded lattice A is distributive if and only if there exists a generalized implication ⇒ defined in A satisfying certain conditions, and we study the class of bounded distributive lattices A endowed with a generalized implication as a common abstraction of the notions of annihilator (Mandelker, Duke Math J 37:377–386, 1970), Quasi-modal algebras (Celani, Math Bohem 126:721–736, 2001), and weakly Heyting algebras (Celani and Jansana, Math Log Q 51:219–246, 2005). We introduce the suitable notions of morphisms in order to obtain a category, as well as the corresponding notion of congruence. We develop a Priestley style topological duality for the bounded distributive lattices with a generalized implication. This duality generalizes the duality given in Celani and Jansana (Math Log Q 51:219–246, 2005) for weakly Heyting algebras and the duality given in Celani (Math Bohem 126:721–736, 2001) for Quasi-modal algebras.     

Maxwell equations and the direction of the electromagnetic field

Let M₀ be the Minkowski space, let Λ₂(M₀) be the space of bivectors in M₀, and let G¹ ⊂ Λ₂(M₀) be the manifold of directions of the physical space, consisting of simple bivectors with square −1. A mapping F: U → Λ₂(M₀), U ⊂ ℝ⁴, satisfying the Maxwell equations is regarded as the tensor of an electromagnetic field in vacuum. The field is described on the basis of a special decomposition F = eω + h(*ω), where the mapping ω: U → G¹ is called the direction of the field, and e: U → (0, +∞) and h: U → ℝ are the electric and magnetic coefficients of the field. The Maxwell equations are reformulated in terms of ω, e, and h. Electromagnetic fields whose set of directions is a point or a one-dimensional subset of G¹ are considered. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 329, 2005, pp. 118–146.     

Analysis of orthogonality and of orbits in affine iterated function systems

We introduce a duality for affine iterated function systems (AIFS) which is naturally motivated by group duality in the context of traditional harmonic analysis. Our affine systems yield fractals defined by iteration of contractive affine mappings. We build a duality for such systems by scaling in two directions: fractals in the small by contractive iterations, and fractals in the large by recursion involving iteration of an expansive matrix. By a fractal in the small we mean a compact attractor X supporting Hutchinson’s canonical measure μ, and we ask when μ is a spectral measure, i.e., when the Hilbert space has an orthonormal basis (ONB) of exponentials . We further introduce a Fourier duality using a matched pair of such affine systems. Using next certain extreme cycles, and positive powers of the expansive matrix we build fractals in the large which are modeled on lacunary Fourier series and which serve as spectra for X. Our two main results offer simple geometric conditions allowing us to decide when the fractal in the large is a spectrum for X. Our results in turn are illustrated with concrete Sierpinski like fractals in dimensions 2 and 3. Research supported in part by the National Science Foundation DMS 0457491.     

Complex Lorentz rotations on vectors⊕pseudovectors

We study the complex structure of the space of vectors and pseudovectors Λ≡ ≡ Λ₁ ⊕ Λ₃ arising from the properties of the Hodge duality in 4-dimensional spacetimes. This structure appears naturally in the framework of its real Clifford geometric algebraCl _1,3 orCl _3,1, in which the Hodge duality is the simple multiplication by the volume unit ∈. Interpreting the linear combination of a scalar and a pseudoscalar α+β∈ as a complex number (Λ₀ ⊕ Λ₄ ⋍ ℂ), the odd subspace Λ_ ∈Cl is a left/right complex linear space, previously studied by Sobczyk. From the real metric definingCl, Λ_ inherits a natural complex bilinear normN(a). Corresponding to this norm there is the group of complex Lorentz rotations Spin_1,3 (ℂ), for which we find a formulation using exclusively the real algebraCl. We discuss its behaviour in some examples and find expressions for different decompositions into real and selfdual, real and anti-selfdual, and real and pure imaginary angle rotations. We finally apply our results to the implementation of the rotations corresponding to Euclidean and null signatures inside the real Lorentzian Clifford algebraCl.     

Schur–Weyl duality over finite fields

We prove a version of the Schur–Weyl duality over finite fields. We prove that for any field k, if k has at least r + 1 elements, the Schur–Weyl duality holds for the rth tensor power of a finite dimensional vector space V. Moreover, if the dimension of V is at least r + 1, the natural map ${{k\mathfrak{S}_r \to \mathsf{End}_{{\rm GL}(V)}(V^{\otimes r})}}We prove a version of the Schur–Weyl duality over finite fields. We prove that for any field k, if k has at least r + 1 elements, the Schur–Weyl duality holds for the rth tensor power of a finite dimensional vector space V. Moreover, if the dimension of V is at least r + 1, the natural map k\mathfrakS_r ? End_GL(V)(V^?r){{k\mathfrak{S}_r \to \mathsf{End}_{{\rm GL}(V)}(V^{\otimes r})}} is an isomorphism. This isomorphism may fail if dim _k V is not strictly larger than r.     

Vapnik-Chervonenkis dimension and (pseudo-)hyperplane arrangements

An arrangement of oriented pseudohyperplanes in affined-space defines on its setX of pseudohyperplanes a set system (or range space) (X, ℛ), ℛ ⊑ 2 ^x of VC-dimensiond in a natural way: to every cellc in the arrangement assign the subset of pseudohyperplanes havingc on their positive side, and let ℛ be the collection of all these subsets. We investigate and characterize the range spaces corresponding tosimple arrangements of pseudohyperplanes in this way; such range spaces are calledpseudogeometric, and they have the property that the cardinality of ℛ is maximum for the given VC-dimension. In general, such range spaces are calledmaximum, and we show that the number of rangesR∈ℛ for whichX - R∈ℛ also, determines whether a maximum range space is pseudogeometric. Two other characterizations go via a simple duality concept and “small” subspaces. The correspondence to arrangements is obtained indirectly via a new characterization of uniforom oriented matroids: a range space (X, ℛ) naturally corresponds to a uniform oriented matroid of rank |X|—d if and only if its VC-dimension isd,R∈ℛ impliesX - R∈ℛ, and |ℛ| is maximum under these conditions. Part of this work was done while the first author was a member of the Graduiertenkolleg “Algorithmische Diskrete Mathematik,” supported by the Deutsche Forschungsgemeinschaft, Grant We 1265/2-1. Part of this work has been supported by the German-Israeli Foundation for Scientific Research and Development (G.I.F.).     

Duality in Function Spaces

In this paper we use Bartle’s technique to study duality between a topological space and a function space. Normally such a duality forms an essential part of Functional Analysis. We introduce several new topologies such as the topology of even convergence T_e, the closed-cocompact topology T_{k ′}, the (strong) local proximal convergence. We explore the topological groups of self-homeomorphisms of a topological space and shed light on the earlier work of Arens, Dieudonné, Di Concilio. We also study the concepts such as evenly equidistant, functionally equicontinuous, due to Bouziad-Troallic and topologically equicontinuous due to Royden. In memory of Professor Enrico Meccariello who made a considerable contribution to this work and who suddenly passed away before his time     

Finite-Range Electromagnetic Interaction and Magnetic Charges Spacetime Algebra or Algebra of Physical Space?

A finite-range electromagnetic (EM) theory containing both electric and magnetic charges constructed using two vector potentials A^μ and Z^μ is formulated in the spacetime algebra (STA) and in the algebra of the three-dimensional physical space (APS) formalisms. Lorentz, local gauge and EM duality invariances are discussed in detail in the APS formalism. Moreover, considerations about signature and dimensionality of spacetime are discussed. Finally, the two formulations are compared. STA and APS are equally powerful in formulating our model, but the presence of a global commuting unit pseudoscalar in the APS formulation and the consequent possibility of providing a geometric interpretation for the imaginary unit employed throughout physics lead us to prefer the APS approach.     

Hermite Constants of Division Algebras

 We introduce a constant attached to a central division algebra D over a number field which is a generalization of the Hermite–Rankin constant. Geometrically, equals the maximum of minimal twisted heights of rational points of a generalized Brauer–Severi variety. We will deduce a duality result, an analog of Rankin’s inequality and an upper estimate for .     

A 51-dimensional embedding of the Ree–Tits generalized octagon

We construct an embedding of the Ree–Tits generalized octagon defined over a field K in a 51-dimensional projective space over K arising from a 52-dimensional Lie algebra J of type . This construction derives from a quadratic map (related to a ‘standard’ duality of ) from the 26-dimensional module (see K. Coolsaet, Adv Geometry, to appear) into J. (This embedding is full if and only if K is a perfect field.) We provide explicit formulas for the coordinates of the points of the octagon in this embedding, in terms of their Van Maldeghem coordinates. We apply these results to compute the dimensions of subspaces generated by various special subsets of points of the octagon: the sets of points at a fixed distance from a given point or a given line and the Suzuki suboctagons. The results depend on whether K is the field of 2 elements, or not.      

Duality Questions for Operators,Spectrum and Measures

We explore spectral duality in the context of measures in ℝ ⁿ , starting with partial differential operators and Fuglede’s question (1974) about the relationship between orthogonal bases of complex exponentials in L ²(Ω) and tiling properties of Ω, then continuing with affine iterated function systems. We review results in the literature from 1974 up to the present, and we relate them to a general framework for spectral duality for pairs of Borel measures in ℝ ⁿ , formulated first by Jorgensen and Pedersen.     

Hermite Constants of Division Algebras

 We introduce a constant attached to a central division algebra D over a number field which is a generalization of the Hermite–Rankin constant. Geometrically, equals the maximum of minimal twisted heights of rational points of a generalized Brauer–Severi variety. We will deduce a duality result, an analog of Rankin’s inequality and an upper estimate for . Received 23 October 2000; in revised form 8 September 2001     

Excessive kernels and Revuz measures

We consider a proper submarkovian resolvent of kernels on a Lusin measurable space and a given excessive measure ξ. With every quasi bounded excessive function we associate an excessive kernel and the corresponding Revuz measure. Every finite measure charging no ξ–polar set is such a Revuz measure, provided the hypothesis (B) of Hunt holds. Under a weak duality hypothesis, we prove the Revuz formula and characterize the quasi boundedness and the regularity in terms of Revuz measures. We improve results of Azéma [2] and Getoor and Sharpe [20] for the natural additive functionals of a Borel right process. Received: 30 April 1997 / Revised version: 17 September 1999 /?Published online: 11 April 2000     

Adiabatic Vacuum States on General Spacetime Manifolds: Definition, Construction, and Physical Properties

Adiabatic vacuum states are a well-known class of physical states for linear quantum fields on Robertson-Walker spacetimes. We extend the definition of adiabatic vacua to general spacetime manifolds by using the notion of the Sobolev wavefront set. This definition is also applicable to interacting field theories. Hadamard states form a special subclass of the adiabatic vacua. We analyze physical properties of adiabatic vacuum representations of the Klein-Gordon field on globally hyperbolic spacetime manifolds (factoriality, quasiequivalence, local definiteness, Haag duality) and construct them explicitly, if the manifold has a compact Cauchy surface.     

Extensions of the duality between minimal surfaces and maximal surfaces

As a generalization of the classical duality between minimal graphs in E ³ and maximal graphs in L ³, we construct the duality between graphs of constant mean curvature H in Bianchi-Cartan-Vranceanu space E ³(κ, τ) and spacelike graphs of constant mean curvature τ in Lorentzian Bianchi-Cartan-Vranceanu space L ³(κ, H).