A Series of Algebras Generalizing the Octonions and Hurwitz-Radon Identity

We study non-associative twisted group algebras over (\mathbbZ₂)ⁿ{(\mathbb{Z}_2)^n} with cubic twisting functions. We construct a series of algebras that extend the classical algebra of octonions in the same way as the Clifford algebras extend the algebra of quaternions. We study their properties, give several equivalent definitions and prove their uniqueness within some natural assumptions. We then prove a simplicity criterion.     

Quantum Groups GLp,q(2)- and $\mathit{SU}_{q_{1}/q_{2}}(2)$ -Invariant Bosonic Gases: A Comparative Study

We discuss the algebras, representations, and thermodynamics of quantum group bosonic gas models with two different symmetries: GL _p,q (2) and . We establish the nature of the basic numbers which follow from these GL _p,q (2)- and -invariant bosonic algebras. The Fock space representations of both of these quantum group invariant bosonic oscillator algebras are analyzed. It is concisely shown that these two quantum group invariant bosonic particle gases have different algebraic and high-temperature thermo-statistical properties.     

Quark mass dependence of masses and decay constants of the pseudo-Goldstone bosons in QCD

The dependence of the pseudo-scalar meson masses and decay constants on sea and valence quark masses is compared to next-to-leading order (NLO) chiral perturbation theory (ChPT). The numerical simulations with two light dynamical quark flavors are performed with the Wilson quark lattice action at gauge coupling and hopping parameters on a 16⁴ lattice. lattice artifacts are taken into account by applying chiral perturbation theory for the Wilson lattice action. The values of the relevant combinations of Gasser-Leutwyler constants L ₄, L ₅, L ₆ and L ₈ are estimated.Received: 7 July 2003, Published online: 2 October 2003     

Search for lepton-flavor violation at HERA

A search for lepton-flavor-violating interactions and has been performed with the ZEUS detector using the entire HERA I data sample, corresponding to an integrated luminosity of . The data were taken at center-of-mass energies, , of 300 and . No evidence of lepton-flavor violation was found, and constraints were derived on leptoquarks (LQs) that could mediate such interactions. For LQ masses below , limits were set on , where is the coupling of the LQ to an electron and a first-generation quark q₁, and is the branching ratio of the LQ to the final-state lepton (μ or ) and a quark q. For LQ masses much larger than , limits were set on the four-fermion interaction term for LQs that couple to an electron and a quark and to a lepton and a quark , where and are quark generation indices. Some of the limits are also applicable to lepton-flavor-violating processes mediated by squarks in R-Parity-violating supersymmetric models. In some cases, especially when a higher-generation quark is involved and for the process , the ZEUS limits are the most stringent to date. Received: 1 April 2005, Revised: 13 July 2005, Published online: 18 October 2005     

Full Field Algebras

We introduce a notion of full field algebra which is essentially an algebraic formulation of the notion of genus-zero full conformal field theory. For any vertex operator algebras V ^L and V ^R , is naturally a full field algebra and we introduce a notion of full field algebra over . We study the structure of full field algebras over using modules and intertwining operators for V ^L and V ^R . For a simple vertex operator algebra V satisfying certain natural finiteness and reductivity conditions needed for the Verlinde conjecture to hold, we construct a bilinear form on the space of intertwining operators for V and prove the nondegeneracy and other basic properties of this form. The proof of the nondegenracy of the bilinear form depends not only on the theory of intertwining operator algebras but also on the modular invariance for intertwining operator algebras through the use of the results obtained in the proof of the Verlinde conjecture by the first author. Using this nondegenerate bilinear form, we construct a full field algebra over and an invariant bilinear form on this algebra.     

Quark mass dependence of pseudoscalar masses and decay constants on a lattice

Our previous calculations of the sea- and valence-quark mass dependence of the pseudoscalar meson masses and decay constants is repeated on a lattice, which allows for a better determination of the quantities in question. The conclusions are similar as before on the 16⁴ lattice [1]. The two light dynamical quark flavours we simulate have masses in the range m _s /4 < m _u,d < 2m _s /3. The sea quark mass dependence of and is well described by the next-to-leading order (NLO) chiral perturbation theory (ChPT) formulas and clearly shows the presence of chiral logarithms. The valence quark mass dependence requires the presence of NNLO contributions in partially quenched ChPT (PQChPT)--in addition to the NLO terms. The lattice artifacts in these quantities turn out to be small.Received: 19 March 2004, Published online: 24 August 2004     

On the relation of Manin’s quantum plane and quantum Clifford algebras

In a recent work we have shown that quantum Clifford algebras — i.e. Clifford algebras of an arbitrary bilinear form — are closely related to the deformed structures asq-spin groups, Hecke algebras,q-Young operators and deformed tensor products. The question to relate Manin’s approach to quantum Clifford algebras is addressed here. Explicit computations using the CLIFFORD Maple package are exhibited. The meaning of non-commutative geometry is reexamined and interpreted in Clifford algebraic terms. Presented at the 9th Colloquium “Quantum Groups and Integrable Systems”, Prague, 22–24 June 2000.     

On realizations of exterior calculus with d N = 0

We study realizations of the q-exterior calculus with exterior differential d satisfying d ^N = 0, N > 2 on the free associative algebra with one generator and on the generalized Clifford algebras. Analogs of the notions of connection and curvature are discussed in the case of the q-exterior calculus on the generalized Clifford algebra. We show that the q-exterior calculus on the free associative algebra with one generator is related to q-calculus on the braided line.     

Wronski Brackets and the Ferris Wheel

We connect the Bayesian order on classical states to a certain Lie algebra on . This special Lie algebra structure, made precise by an idea we introduce called a Wronski bracket, suggests new phenomena the Bayesian order naturally models. We then study Wronski brackets on associative algebras, and in the commutative case, discover the beautiful result that they are equivalent to derivations. PACS: 03.67.a, 02.20.Sv.     

Effect of top quark spin on the unparticle couplings in $\gamma\gamma \to t\bar{t}$

We investigate the potential of γ γ collisions to probe scalar unparticle couplings via top–antitop quark pair production. We find 95% confidence level limits on the unparticle couplings with an integrated luminosity of 500 fb⁻¹ and an energy of  TeV. We investigate the effect of the top quark spin polarization on the unparticle couplings. It is shown that spin polarization of the top quark leads to a significant improvement in the sensitivity limits.     

ℤ n 3 -graded colored supersymmetry

We build generalizations of the Grassmann algebras from a few following simple assumptions: the algebras are graded, maximally symmetric and contain an ordinary Grassmann algebra as a subalgebra. These algebras are graded by and display surprising properties that indicate their possible application to the modeling of quark fields. We build the generalized supersymmetry generators based on these algebras and their derivation operators. These generators are cubic roots of the usual supersymmetry generators.     

Octonionic representations of Clifford algebras and triality

The theory of representations of Clifford algebras is extended to employ the division algebra of the octonions or Cayley numbers. In particular, questions that arise from the nonassociativity and noncommutativity of this division algebra are answered. Octonionic representations for Clifford algebras lead to a notion of octonionic spinors and are used to give octoninic representations of the respective orthogonal groups. Finally, the triality automorphisms are shown to exhibit a manifest ₃ ×SO(8) structure in this framework.     

An Algebraic Derivation of the Eigenspaces Associated with an Ising-Like Spectrum of the Superintegrable Chiral Potts Model

In terms of the loop algebra and the algebraic Bethe-ansatz method, we derive the invariant subspace associated with a given Ising-like spectrum consisting of 2 ^r eigenvalues of the diagonal-to-diagonal transfer matrix of the superintegrable chiral Potts (SCP) model with arbitrary inhomogeneous parameters. We show that every regular Bethe eigenstate of the τ ₂-model leads to an Ising-like spectrum and is an eigenvector of the SCP transfer matrix which is given by the product of two diagonal-to-diagonal transfer matrices with a constraint on the spectral parameters. We also show in a sector that the τ ₂-model commutes with the loop algebra, , and every regular Bethe state of the τ ₂-model is of highest weight. Thus, from physical assumptions such as the completeness of the Bethe ansatz, it follows in the sector that every regular Bethe state of the τ ₂-model generates an -degenerate eigenspace and it gives the invariant subspace, i.e. the direct sum of the eigenspaces associated with the Ising-like spectrum.     

Electroweak precision constraints on vector-like fermions

We calculate the oblique electroweak corrections and confront them with the experiments in an extension of the standard model. The new fields added are a vector-like weak doublet and a singlet fermion. After electroweak symmetry breaking there is a mixing between the components of the new fields, but there is no mixing allowed with the standard fermions. Four electroweak parameters, , , W and Y, are presented in the formalism of Barbieri et al.; these are the generalization of the Peskin–Takeuchi S, T and U. The vector-like extension is slightly constrained. requires the new neutral fermion masses not to be very far from each other, allowing for higher mass differences for higher masses and smaller mixing. , W and Y give practically no constraints on the masses. This extension can give a positive contribution to , allowing for a heavy Higgs boson in electroweak precision tests of the standard model.     

Peirce, Clifford, and Quantum Theory

Beginning in 1870 Charles Sanders Peirce published a series of papers on a logic of relations, which corresponded to a linear associative algebra. This algebra is related by a linear transformation to quaternions and thus to the C(3, 0) algebra of William Kingdon Clifford. This Clifford algebra contains the Pauli matrices and so constitutes an operator basis for the nonrelativistic quantum theory of spin one-half particles. A further unification is achieved by taking the wave functions themselves to be 2 × 2 matrices which are Peirce logical operators and also elements of the Clifford algebra. Thus we have discovered a direct path from the Peirce logic to quantum theory. A diagrammatic method follows from the Peirce/Clifford algebraic approach and is suitable for describing particle interactions.     

Quantization on Curves

Deformation quantization on varieties with singularities offers perspectives that are not found on manifolds. The Harrison component of Hochschild cohomology, vanishing on smooth manifolds, reflects information about singularities. The Harrison 2-cochains are symmetric and are interpreted in terms of abelian *-products. This paper begins a study of abelian quantization on plane curves over , being algebraic varieties of the form , where R is a polynomial in two variables; that is, abelian deformations of the coordinate algebra ). To understand the connection between the singularities of a variety and cohomology we determine the algebraic Hochschild (co)homology and its Barr–Gerstenhaber–Schack decomposition. Homology is the same for all plane curves , but the cohomology depends on the local algebra of the singularity of R at the origin. The Appendix, by Maxim Kontsevich, explains in modern mathematical language a way to calculate Hochschild and Harrison cohomology groups for algebras of functions on singular planar curves etc. based on Koszul resolutions.