Ternary Virasoro–Witt algebra

A 3-bracket variant of the Virasoro–Witt algebra is constructed through the use of su(1,1)$\mathit{su}(1,1)$ enveloping algebra techniques. The Leibniz rules for 3-brackets acting on other 3-brackets in the algebra are discussed and verified in various situations.     

One-operation image algebra for optical processing

A binary image algebra with one operation of a logic operation followed by a dilation and its modification with three additional operations for threshold processing of grey level images are developed in this paper. All the binary image processing functions and various grey level image processing functions related to morphological operations can be represented by the algebraic structure. The algebra is particularly suitable for parallel processing by optics in a cellular logic image processor architecture.     

Fast algebra algorithm of shape-from-shading with specular reflectance

Shape-from-shading (SFS) is to reconstruct three-dimensionai (3D) shape from a single gray image,which is an important problem in computer vision.We propose a novel SFS method based on hybrid reflection model which contains both diffuse reflectance and specular reflectance.The intensity gradient of image is in the direction that the shape of surface changes most,so we use directional derivative of the reflectance map as parts of objective function.When discrete characteristic of digital images is considered,finite difference approximates differential operator.So the reflectance map equation described by a partial differential equation (PDE) turns into an algebra equation about the unknown surface height correspondingly.Using iterative numeric computation,a new SFS method is gained.Experiments on synthesis and real images show that the proposed SFS method is accurate and fast.     

TrigonometricR matrices related to ‘dilute’ Birman-Wenzl-Murakami algebra

Explicit expressions for three series ofR matrices which are related to a dilute generalisation of the Birman-Wenzl-Murakami algebra are presented. Of those, one series is equivalent to the quantumR matrices of theD _n+1 ⁽²⁾ generalised Toda systems, whereas the remaining two series appear to be new.     

Current algebra Ward identities in the renormalized σ model

The many-current Ward identities corresponding to the Gell-Mann current algebra are discussed in the renormalized model. The Ward identities are verified in the case of the SU(2)×SU(2) chiral symmetry. In the SU(3)×SU(3) case the uniqueness of the Adler-Bardeen anomaly is proved using the Wess-Zumino consistency conditions.On leave of absence from the University of Genova (Italy).Chercheur Associé au C.N.R.S. (C.P.T./Marseille).     

C λ-extended oscillator algebra and parasupersymmetric quantum mechanics

The C -extended oscillator algebra is generated by {1, a, a , N, T}, where T is the generator of the cyclic group C of order . It can be realized as a generalized deformed oscillator algebra (GDOA). Its unirreps can thus be easily exhibited using the representation theory of GDOAs and their carrier spaces show a Z-grading structure. Within its infinite-dimensional Fock space representation, this algebra provides a bosonization of parasupersymmetric quantum mechanics of order p = – 1.     

Noncoassociative twisting: Its application to κ-Poincaré algebra

It is shown that the use of noncoassociative coproduct allows us to simplify the structure of-Poincaré algebra — it contains nondeformedD=3 Euclidean quasi-bialgebra. We obtain by the dual construction the commuting, nonassociativeD=4 space-time.Presented at the 4th international Colloquium Quantum Groups and Integrable Systems, Prague, 22–24 June, 1995I would like to thank Profs. J. Lukierski and V.N. Tolstoy for valuable comments.     

Quasifinite highest weight modules over the superW 1+∞algebra

We study quasifinite highest weight modules over the supersymmetric extension of theW ₁₊ algebra on the basis of the analysis by Kac and Radul. We find that the quasifiniteness of the modules is again characterized by polynomials, and obtain the differential equations for highest weights. The spectral flow, free field realization over the (B, C)-system, and the embedding into (|) are also presented.Address after April 1, 1994: Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606, JapanAddress after April 1, 1994: Uji Research Center, Yukawa Institute for Theoretical Physics, Kyoto University, Uji 611, Japan     

Imaginary numbers are not real—The geometric algebra of spacetime

This paper contains a tutorial introduction to the ideas of geometric algebra, concentrating on its physical applications. We show how the definition of a geometric product of vectors in 2-and 3-dimensional space provides precise geometrical interpretations of the imaginary numbers often used in conventional methods. Reflections and rotations are analyzed in terms of bilinear spinor transformations, and are then related to the theory of analytic functions and their natural extension in more than two dimensions (monogenics), Physics is greatly facilitated by the use of Hestenes' spacetime algebra, which automatically incorporates the geometric structure of spacetime. This is demonstrated by examples from electromagnetism. In the course of this purely classical exposition many surprising results are obtained—results which are usually thought to belong to the preserve of quantum theory. We conclude that geometric algebra is the most powerful and general language available for the development of mathematical physics.The title of this paper is inspired by David Hestenes, who is known to have a fondness for deliberate ambiguity.⁽¹⁾ Supported by a SERC studentship.     

Quasi-integrable and superintegrable systems from a ‘deformed-mass’ two-photon algebra

A quantum deformation of the two-photon (or Schrödinger) Lie algebra is introduced in order to construct newn-dimensional classical Hamiltonian systems which have (n?2) functionally independent integrals of motion in involution; we say that such Hamiltonians define quasi-integrable systems. Furthermore, Hopf subalgebras of this quantum two-photon algebra (quantum extended Galilei and harmonic oscillator algebras) provide another set of (n?1) integrals of motion for Hamiltonians defined on these Hopf subalgebras, so that they lead to superintegrable systems.     

A 9 × 9 matrix representation of Temperleyben-Lieb algebra and corresponding entanglement

We present the matrix U. The 9 × 9 U matrix is a representation of specialized Temperley-Lieb algebra. Based on which, a unitary Ř matrix is generated via the Yang-Baxterization approach. The 9 × 9 Ř matrix, a solution of the Yang-Baxter equation, is universal for quantum entanglement.     

Character and determinant formulae of quasifinite representation of theW 1+∞ algebra

We diagonalize the Hilbert space of some subclass of the quasifinite module of theW ₁₊ algebra. States are classified according to their eigenvalues for infinitely many commuting charges and the Young diagrams. The parameter dependence of their norms is explicitly derived. The full character formulae of the degenerate representations are given as summation of the bilinear combinations of the Schur polynomials.     

A subalgebra of loop algebra A2^～ and its applications

A subalgebra of loop algebra A2^～ and its expanding loop algebra G^- are constructed. It follows that both resulting integrable Hamiltonian hierarchies are obtained. As a reduction case of the first hierarchy, a generalized nonlinear coupled Schroedinger equation, the standard heat-conduction and a formalism of the well known Ablowitz, Kaup, Newell and Segur hierarchy are given, respectively. As a reduction case of the second hierarchy, the nonlinear Schroedinger and modified Korteweg de Vries hierarchy and a new integrable system are presented. Especially, a coupled generalized Burgers equation is generated.     

Beyond the operator algebra—To be nonlocal realism of quantum mechanics

正In 1905,Albert Einstein presented his milestone theory of photons in his paper titled "On a Heuristic Viewpoint Concerning the Production and Transformation of Light"based on Max Planck’s black-body radiation theory.The groundbreaking theory successfully solved the paradox between Maxwell’s wave theory of light and the experimental finding     

A 9 × 9 matrix representation of Temperley-Lieb algebra and corresponding entanglement

We present the matrix U.The 9×9 U matrix is a representation of specialized Temperley-Lieb algebra.Based on which,a unitary R matrix is generated via the Yang-Baxterization approach.The 9×9 R matrix,a solution of the Yang-Baxter equation,is universal for quantum entanglement.     

D=2 and D=4 realization of κ-conformal algebra

We describe the generators of-conformal transformations leaving invariant the-deformed d'Alembert equation. For the case D=4 the algebraic structure of the conformal extension of the off-shell spin zero realization of-Poincaré algebra is discussed. Then the D=2 off-shell realization of-conformal algebra for arbitrary spin and its commutation relations are studied.Presented at the 4th Colloquium Quantum Groups and Integrable Systems, Prague, 22–24 June 1995.Supported by KBN grant 2P 302 087 06.     

Differential calculus onISO q(N), quantum Poincaré algebra and q-gravity

We present a general method to deform the inhomogeneous algebras of theB _n,C_n,D_n type, and find the corresponding bicovariant differential calculus. The method is based on a projection fromB _n+1,C_n+1,D_n+1. For example we obtain the (bicovariant) inhomogeneousq-algebraISO _q(N) as a consistent projection of the (bicovariant)q-algebraSO _q(N=2). This projection works for particular multiparametric deformations ofSO(N+2), the so-called minimal deformations. The case ofISO _q(4) is studied in detail: a real form corresponding to a Lorentz signature exists only for one of the minimal deformations, depending on one parameterq. The quantum Poincaré Lie algebra is given explicitly: it has 10 generators (no dilatations) and contains theclassical Lorentz algebra. Only the commutation relations involving the momenta depend onq. Finally, we discuss aq-deformation of gravity based on the gauging of thisq-Poincaré algebra: the lagrangian generalizes the usual Einstein-Cartan lagrangian.     

Generalized Klein–Gordon field equations with octonion space-time (OST) algebra

The concept of space-time representation is redefined using the octonion space-time (OST) algebra. In this study, describing the properties of octonions and their possible connection with Euclidean space-times, the internal and external space-time events are represented within the OST algebra. Keeping in mind the octonionic dual-Euclidean space-times, we express the homogeneous field equations which leads to the symmetrical nature of internal and external space-times. We derive the generalized Proca–Maxwell equations for massive-dyons in the case of the OST algebra. Accordingly, we have obtained a new set of octonionic Klein–Gordon potential (KGP) and Klein–Gordon field (KGF) equations for massive dyons from the generalized Proca–Maxwell equations. This formalism demonstrates that the octonionic KGP and KGF equations can be expressed in a single equation and it is equivalent to energy-momentum relation for dyons. As such, we have made an attempt to write the conservation of Noetherian current from the octonionic Klein–Gordon equations.