Positive Principal Minor Property of Linear Transformations on Euclidean Jordan Algebras

In this article, we study the positive principal minor (PPM) property of linear transformations on Euclidean Jordan algebras. Specifically, we give a characterization of the PPM property on the Lorentz space ℒ ⁿ and show that the PPM property implies the Q property. We also study a matrix-induced transformation on ℒ ⁿ . The author is grateful to Professor M. S. Gowda for helpful suggestions.     

Quantum Fractals on <Emphasis Type="Italic">n</Emphasis>–Spheres. Clifford Algebra Approach

Using the Clifford algebra formalism we extend the quantum jumps algorithm of the Event Enhanced Quantum Theory (EEQT) to convex state figures other than those stemming from convex hulls of complex projective spaces that form the basis for the standard quantum theory. We study quantum jumps on n-dimensional spheres, jumps that are induced by symmetric configurations of non-commuting state monitoring detectors. The detectors cause quantum jumps via geometrically induced conformal maps (M?bius transformations) and realize iterated function systems (IFS) with fractal attractors located on n-dimensional spheres. We also extend the formalism to mixed states, represented by “density matrices” in the standard formalism, (the n-balls), but such an extension does not lead to new results, as there is a natural mechanism of purification of states. As a numerical illustration we study quantum fractals on the circle (one-dimensional sphere and pentagon), two–sphere (octahedron), and on three-dimensional sphere (hypercubetesseract, 24 cell, 600 cell, and 120 cell). The attractor, and the invariant measure on the attractor, are approximated by the powers of the Markov operator. In the appendices we calculate the Radon-Nikodym derivative of the SO(n + 1) invariant measure on Sⁿ under SO(1, n + 1) transformations and discuss the Hamilton’s “icossian calculus” as well as its application to quaternionic realization of the binary icosahedral group that is at the basis of the 600 cell and its dual, the 120 cell. As a by-product of this work we obtain several Clifford algebraic results, such as a characterization of positive elements in a Clifford algebra as generalized Lorentz “spin–boosts”, and their action as M?bius transformation on n-sphere, and a decomposition of any element of Spin⁺(1, n + 1) into a spin–boost and a spin–rotation, including the explicit formula for the pullback of the SO(n + 1) invariant Riemannian metric with respect to the associated M?bius transformation.     

Hyperbolic Calculus

The complex numbers are naturally related to rotations and dilatations in the plane. In this paper we present the function theory associate to the (universal) Clifford algebra forIR ^1,0 [1], the so called hyperbolic numbers [2,3,4], which can be related to Lorentz transformations and dilatations in the two dimensional Minkowski space-time. After some brief algebraic interpretations (part 1), we present a “Hyperbolic Calculus” analogous to the “Calculus of one Complex Variable”. The hyperbolic Cauchy-Riemann conditions, hyperbolic derivatives and hyperbolic integrals are introduced on parts 2 and 3. Then special emphasis is given in parts 4 and 5 to conformal hyperbolic transformations which preserve the wave equation, and hyperbolic Riemann surfaces which are naturally associated to classical string motions.     

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.     

Rank 2 Nichols Algebras with Finite Arithmetic Root System

The concept of arithmetic root systems is introduced. It is shown that there is a one-to-one correspondence between arithmetic root systems and Nichols algebras of diagonal type having a finite set of (restricted) Poincaré–Birkhoff–Witt generators. This has strong consequences for both objects. As an application all rank 2 Nichols algebras of diagonal type having a finite set of (restricted) Poincaré–Birkhoff–Witt generators are determined. Supported by the European Community under a Marie Curie Intra-European Fellowship.     

Cleft Extensions of Hopf Algebroids

The notions of a cleft extension and a cross product with a Hopf algebroid are introduced and studied. In particular it is shown that an extension (with a Hopf algebroid ℋ = (ℋ _L , ℋ _R )) is cleft if and only if it is ℋ _R -Galois and has a normal basis property relative to the base ring L of ℋ _L . Cleft extensions are identified as crossed products with invertible cocycles. The relationship between the equivalence classes of crossed products and gauge transformations is established. Strong connections in cleft extensions are classified and sufficient conditions are derived for the Chern–Galois characters to be independent on the choice of strong connections. The results concerning cleft extensions and crossed product are then extended to the case of weak cleft extensions of Hopf algebroids hereby defined. Dedicated to Stef Caenepeel on the occasion of his 50th birthday.     

On matrix transformations and sequence spaces

In this paper are given results on the spacesw _τ (μ) andc _τ (μ, μ′) the second one generalizing the well-known spacec _∞ (μ) of sequences that are strongly bounded. Then we deal with matrix transformations into these spaces. These results generalize those given in [7].     

Faraday’s Form and Maxwell’s Equations in the Heisenberg Group

In this note we present a geometric formulation of Maxwell’s equations in Carnot groups (connected simply connected nilpotent Lie groups with stratified Lie algebra) in the setting of the intrinsic complex of differential forms defined by M. Rumin. Restricting ourselves to the first Heisenberg group \mathbbH¹{\mathbb{H}^{1}}, we show that these equations are invariant under the action of suitably defined Lorentz transformations, and we prove the equivalence of these equations with differential equations “in coordinates”. Moreover, we analyze the notion of “vector potential”, and we show that it satisfies a new class of 4th order evolution differential equations.     

Symmetries,Lagrangian and Conservation Laws for the Maxwell Equations

The evolution equations of Maxwell’s equations has a Lagrangian written in terms of the electric E and magnetic H fields, but admit neither Lorentz nor conformal transformations. The additional equations ∇⋅E=0, ∇⋅H=0 guarantee the Lorentz and conformal invariance, but the resulting system is overdetermined, and hence does not have a Lagrangian. The aim of the present paper is to attain a harmony between these two contradictory properties and provide a correspondence between symmetries and conservation laws using the Lagrangian for the evolutionary part of Maxwell’s equations.     

On Dynamics Related to a Class of Numeration Systems

 In this paper, we investigate the class of numeration systems and we study the associated dynamical systems, called odometers. It is shown that these odometers are measure-theoretically isomorphic to rank one transformations on the unit interval, constructed by a cutting-stacking method. Furthermore, a symbolic coding leads to isomorphic shift systems arising from substitutions. Some skew products of the odometers by cocycles related to the sum of digits are shown to be ergodic.     

Integration of Categorical Frameworks: Rule-Based Refinement and Hierarchical Composition for Components

The integration of two important categorical frameworks – namely adhesive High-Level Replacement (HLR) systems and the generic component concept– yields a categorical approach to component transformation and refinement. The generic component concept is shown to be an adhesive HLR category, so rules and transformations as well as the corresponding results are available. Moreover, the compatibility with the hierarchical component composition is provided. The extension to rule-based refinement requires additional property-preserving morphisms and yields property-preserving rules and transformations, i.e. refinements where compatibility with the hierarchical component composition again is achieved. The categorical framework is instantiated to typed algebraic high-level (AHL) nets and illustrated with an example of AHL net components.      

Clifford algebra and the Lorentz transformation with (p, q) form

In this paper, the concepts of Lorentz inner product with (p, q) form, the Lorentz space and the Lorentz transformation with (p, q) form are given by using Clifford algebra. It is shown that L_m^p,q is the Lorentz transformation with (p, q) form, and the matrix equality relation of Minkowski space with (n − 1, 1) form is given. The examples are given to illustrate the corresponding results.     

On Dynamics Related to a Class of Numeration Systems

 In this paper, we investigate the class of numeration systems and we study the associated dynamical systems, called odometers. It is shown that these odometers are measure-theoretically isomorphic to rank one transformations on the unit interval, constructed by a cutting-stacking method. Furthermore, a symbolic coding leads to isomorphic shift systems arising from substitutions. Some skew products of the odometers by cocycles related to the sum of digits are shown to be ergodic. Received 5 March 2001; in revised form 16 August 2001     

Limits of rank 4 Azumaya algebras and applications to desingularization

It is shown that the schematic image of the scheme of Azumaya algebra structures on a vector bundle of rank 4 over any base scheme is separated, of finite type, smooth of relative dimension 13 and geometrically irreducible over that base and that this construction base-changes well. This fully generalizes Seshadri’s theorem in [16] that the variety of specializations of (2 x 2)-matrix algebras is smooth in characteristic ≠ 2. As an application, a construction of Seshadri in [16] is shown in a characteristic-free way to desingularize the moduli space of rank 2 even degree semi-stable vector bundles on a complete curve. As another application, a construction of Nori over ℤ (Appendix, [16]) is extended to the case of a normal domain which is a universally Japanese (Nagata) ring and is shown to desingularize the Artin moduli space [1] of invariants of several matrices in rank 2. This desingularization is shown to have a good specialization property if the Artin moduli space has geometrically reduced fibers — for example this happens over ℤ. Essential use is made of Kneser’s concept [8] of ‘semi-regular quadratic module’. For any free quadratic module of odd rank, a formula linking the half-discriminant and the values of the quadratic form on its radical is derived.     

Representations of deformed preprojective algebras and quantum groups

Let (Γ,I) be the bound quiver of a cyclic quiver whose vertices correspond to the Abelian group Zd. In this paper, we list all indecomposable representations of (Γ,I) and give the conditions that those representations of them can be extended to representations of deformed preprojective algebra Πλ(Γ,I). It is shown that those representations given by extending indecomposable representations of (Γ,I) are all simple representations of Πλ(Γ,I). Therefore, it is concluded that all simple representa-tions of rest...     

Strict Semimonotonicity Property of Linear Transformations on Euclidean Jordan Algebras

Motivated by the equivalence of the strict semimonotonicity property of the matrix A and the uniqueness of the solution to the linear complementarity problem LCP(A,q) for q∈R ₊ ⁿ , we study the strict semimonotonicity (SSM) property of linear transformations on Euclidean Jordan algebras. Specifically, we show that, under the copositive condition, the SSM property is equivalent to the uniqueness of the solution to LCP(L,q) for all q in the symmetric cone K. We give a characterization of the uniqueness of the solution to LCP(L,q) for a Z transformation on the Lorentz cone ℒ₊ ⁿ . We study also a matrix-induced transformation on the Lorentz space ℒ ⁿ .     

Complete classification of parallel Lorentz surfaces in neutral pseudo hyperbolic 4-space

A Lorentz surface of an indefinite space form is called a parallel surface if its second fundamental form is parallel with respect to the Van der Waerden-Bortolotti connection. Such surfaces are locally invariant under the reflection with respect to the normal space at each point. Parallel surfaces are important in geometry as well as in general relativity since extrinsic invariants of such surfaces do not change from point to point. Recently, parallel Lorentz surfaces in 4D neutral pseudo Euclidean 4-space $ \mathbb{E}_2^4 $ \mathbb{E}_2^4 and in neutral pseudo 4-sphere S ₂⁴ (1) were classified in [14] and in [10], respectively. In this paper, we completely classify parallel Lorentz surfaces in neutral pseudo hyperbolic 4-space H ₂⁴ (−1). Our main result states that there are 53 families of parallel Lorentz surfaces in H ₂⁴ (−1). Conversely, every parallel Lorentz surface in H ₂⁴ (−1) is obtained from the 53 families. As an immediate by-product, we achieve the complete classification of all parallel Lorentz surfaces in 4D neutral indefinite space forms.     

The possibility of relativistic Finslerian geometry

Foundations of Finslerian geometry that are of interest for solving the problem of geometrization of classical electrodynamics in metric four-dimensionality are investigated. It is shown that parametrization of the interval—the basic aspect of geometry—is carried out non-relativistically. A relativistic way of parametrization is suggested, and the corresponding variant of the geometry is constructed. The equation for the geodesic of this variant of geometry, aside from the Riemannian, has a generalized Lorentz term, the connection contains an additional Lorentz tensorial summand, and the first schouten is different from zero. Some physical consequences of the new geometry are considered: the non-measurability of the generalized electromagnetic potential in the classical case and its measurability on quantum scales (the Aharonov-Bohm effect); it is shown that in the quantum limit the hypothesis of discreteness of space-time is plausible. The linear effect with respect to the field of the “redshift” is also considered and contemporary experimental possibilities of its registration are estimated; it is shown that the experimental results could uniquely determine the choice between the standard Riemannian and relativistic Finslerian models of space-time. __________ Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 22, Geometry, 2007.     

Topology and a lorentz-invariant pseudo-riemannian metric of the manifold of directions in the physical space

In the mathematical model of the special relativity theory, a two-dimensional Minkowski subspace is treated as a one-dimensional direction in the physical space. The manifold of such planes is naturally endowed with the structure of a pseudo-Riemannian manifold on which the group of isochronous Lorentz transformations acts transitively by isometries. In this paper, the topology and the metric geometry of this manifold are studied. Bibliography: 4 titles. Translated from Zapiski Nauchnykh Seminar POMI, Vol. 246, 1997, pp. 141–151. Translated by S. Yu. Pilyugin.     

Weak Tensor Category and Related Generalized Hopf Algebras

There are at least two kinds of generalization of Hopf algebra, i.e. pre-Hopf algebra and weak Hopf algebra. Correspondingly, we have two kinds of generalized bialgebras, almost bialgebra and weak bialgebra. Let L = （L, ×, I, a, l, r） be a tensor category. By giving up I, l, r and keeping ×, a in L, the first author got so-called pre-tensor category L = （L, ×, a） and used it to characterize almost bialgebra and pre-Hopf algebra in Comm. in Algebra, 32（2）： 397-441 （2004）. Our aim in this paper is to generalize tensor category L = （L, ×, I, a, l, r） by weakening the natural isomorphisms l, r, i.e. exchanging the natural isomorphism ll^-1 = rr^-1 = id into regular natural transformations lll= l, rrr = r with some other conditions and get so-called weak tensor category so as to characterize weak bialgebra and weak Hopf algebra. The relations between these generalized （bialgebras） Hopf algebras and two kinds generalized tensor categories will be described by using of diagrams. Moreover, some related concepts and properties about weak tensor category will be discussed.