On Nested Bethe Ansatz for RTT Algebra of o(2n) Type

We study the highest weight representations of the RTT algebras for the R matrix of o(2n) type by the nested algebraic Bethe ansatz. We show how auxiliary RTT algebra à can be used to find Bethe vectors and Bethe conditions. For special representations, in which representation of RTT algebra à is trivial, the problem was solved by Reshetikhin.

     

Finite-dimensional representations of the quantum superalgebraU q[gl(n/m)] and relatedq-identities

Explicit expressions for the generators of the quantum superalgebraU _q [gl(n/m)] acting on a class of irreducible representations are given. The class under consideration consists of all essentially typical representations: for these a Gel'fand-Zetlin basis is known. The verification of the quantum superalgebra relations to be satisfied is shown to reduce to a set ofq-number identities.     

Compact matrix pseudogroups

The compact matrix pseudogroup is a non-commutative compact space endowed with a group structure. The precise definition is given and a number of examples is presented. Among them we have compact group of matrices, duals of discrete groups and twisted (deformed)SU(N) groups. The representation theory is developed. It turns out that the tensor product of representations depends essentially on their order. The existence and the uniqueness of the Haar measure is proved and the orthonormality relations for matrix elements of irreducible representations are derived. The form of these relations differs from that in the group case. This is due to the fact that the Haar measure on pseudogroups is not central in general. The corresponding modular properties are discussed. The Haar measures on the twistedSU(2) group and on the finite matrix pseudogroup are found.     

Quantum Charges and Spacetime Topology: The Emergence of New Superselection Sectors

A new form of superselection sectors of topological origin is developed. By that it is meant a new investigation that includes several extensions of the traditional framework of Doplicher, Haag and Roberts in local quantum theories. At first we generalize the notion of representations of nets of C*–algebras, then we provide a brand new view on selection criteria by adopting one with a strong topological flavour. We prove that it is coherent with the older point of view, hence a clue to a genuine extension. In this light, we extend Roberts’ cohomological analysis to the case where 1–cocycles bear non-trivial unitary representations of the fundamental group of the spacetime, equivalently of its Cauchy surface in the case of global hyperbolicity. A crucial tool is a notion of group von Neumann algebras generated by the 1–cocycles evaluated on loops over fixed regions. One proves that these group von Neumann algebras are localized at the bounded region where loops start and end and to be factorial of finite type I. All that amounts to a new invariant, in a topological sense, which can be defined as the dimension of the factor. We prove that any 1–cocycle can be factorized into a part that contains only the charge content and another where only the topological information is stored. This second part much resembles what in literature is known as geometric phases. Indeed, by the very geometrical origin of the 1–cocycles that we discuss in the paper, they are essential tools in the theory of net bundles, and the topological part is related to their holonomy content. At the end we prove the existence of net representations. Dedicated to Klaus Fredenhagen on the occasion of his sixtieth birthday     

Nuclear Algebraic Geometry and SO(10)

In this contribution nuclear representations of the Dirac ring, developed over many years, are shown to be a particular case of a theorem in algebraic geometry which at the same time associates them with a Hodge decomposition of a Kaehler manifold. This yields a shape that in some cases is independent of any appeal to a symmetry group. However, because the nuclear representations are in the infinitesimal ring of SO(4) and the internal space of each representation is in a Kaehler (even Calabi-Yau) manifold K; the group SO(10) = SO(4) × K can give additional information. This paper develops the very fruitful symbiosis between algebra and irreducible representations of SO(10) and covers some aspects of string theory.     

Finite-dimensional representations of U q (osp(1/2n)) and its connection with quantum so(2n+1)

It is shown that the quantum supergroup U _q (osp(1/2_n)) is essentially isomorphic to the quantum group U _-q (so(2n+1)) restricted to tensorial representations. This renders it straightforward to classify all the finite-dimensional irreducible representations of U _q (osp(1/2n)) at generic q. In particular, it is proved that at generic q, every-dimensional irrep of this quantum supergroup is a deformation of an osp(1/2n) irrep, and all the finite-dimensional representations are completely reducible.     

Stochastic Dynamics of Discrete Curves and Multi-Type Exclusion Processes

This study deals with continuous limits of interacting one-dimensional diffusive systems, arising from stochastic distortions of discrete curves with various kinds of coding representations. These systems are essentially of a reaction-diffusion nature. In the non-reversible case, the invariant measure has in general a non Gibbs form. The corresponding steady-state regime is analyzed in detail, by using a tagged particle together with a state-graph cycle expansion of the probability currents. As a consequence, the constants appearing in Lotka–Volterra equations—which describe the fluid limits of stationary states—can be traced back directly at the discrete level to tagged particle cycles coefficients. Current fluctuations are also studied and the Lagrangian is obtained via an iterative scheme. The related Hamilton–Jacobi equation, which leads to the large deviation functional, is investigated and solved in the reversible case, just for the sake of checking.     

Symmetries of quantum spaces. Subgroups and quotient spaces of quantumSU(2) andSO(3) groups

We prove that each action of a compact matrix quantum group on a compact quantum space can be decomposed into irreducible representations of the group. We give the formula for the corresponding multiplicities in the case of the quotient quantum spaces. We describe the subgroups and the quotient spaces of quantumSU(2) andSO(3) groups.     

Fermions in Odd Space-Time Dimensions: Back to Basics

It is a well-known feature of odd space-time dimensions d that there exist two inequivalent fundamental representations A and B of the Dirac gamma matrices. Moreover, the parity transformation swaps the fermion fields living in A and B. As a consequence, a parity-invariant Lagrangian can only be constructed by incorporating both the representations. Based upon these ideas and contrary to long-held belief, we show that in addition to a discrete exchange symmetry for the massless case, we can also define chiral symmetry provided the Lagrangian contains fields corresponding to both the inequivalent representations. We also study the transformation properties of the corresponding chiral currents under parity and charge-conjugation operations. We work explicitly in 2 + 1 dimensions and later show how some of these ideas generalize to an arbitrary number of odd dimensions.     

Some aspects of auger electron spectra of 3d transition metal oxides

Auger electron spectra of the transition metals Cr, Mn, Fe, Co and Ni as well as their oxides have been investigated in the energy range between 0–100 eV. In each case of the clean metal surface the observed spectrum consists essentially of one Auger line identified asM _2,3 VV transition. After oxidation a line doublet is observed revealing two transitions instead of one. Additional new Auger peaks appear in the low energy range between 0–30 eV. The “splitting” of the Auger line can be explained as resulting from aM _2,3 V _dV_d and aM _2,3 V _pV_p transition. The latter is characteristic for the compound and can in a simple way be interpreted as a cross transition.     

New solvable lattice models in three dimensions

In this paper we establish a remarkable connection between two seemingly unrelated topics in the area of solvable lattice models. The first is the Zamolodchikov model, which is the only nontrivial model on a three-dimen-sional lattice so far solved. The second is the chiral Potts model on the square lattice and its generalization associated with theU _q(sl(n)) algebra, which is of current interest due to its connections with high-genus algebraic curves and with representations of quantum groups at roots of unity. We show that this last sl(n)-generalized chiral Potts model can be interpreted as a model on a threedimensional simple cubic lattice consisting ofn square-lattice layers with anN- valued (N2) spin at each site. Further, in theN=2 case this three-dimen-sional model reduces (after a modification of the boundary conditions) to the Zamolodchikov model we mentioned above.     

Spinor-vector duality in heterotic string vacua

Classification of the N=1 space–time supersymmetric fermionic Z₂×Z₂ heterotic-string vacua with symmetric internal shifts, revealed a novel spinor-vector duality symmetry over the entire space of vacua, where the S_t↔V duality interchanges the spinor plus anti-spinor representations with vector representations. In this paper we demonstrate that the spinor-vector duality exists also in fermionic Z₂ heterotic string models, which preserve N=2 space–time supersymmetry. In this case the interchange is between spinorial and vectorial representations of the unbroken SO(12) GUT symmetry. We provide a general algebraic proof for the existence of the S_t↔V duality map. We present a novel basis to generate the free fermionic models in which the ten-dimensional gauge degrees of freedom are grouped into four groups of four, each generating an SO(8) modular block. In the new basis the GUT symmetries are produced by generators arising from the trivial and non-trivial sectors, and due to the triality property of the SO(8) representations. Thus, while in the new basis the appearance of GUT symmetries is more cumbersome, it may be more instrumental in revealing the duality symmetries that underly the string vacua.     

Group-Theoretical Approach to Extended Conformal Supersymmetry: Function Space Realizations and Invariant Differential Operators

We give function space realizations of all representations of the conformal superalgebra su(2,2/N) and of the supergroup SU(2, 2 /N) induced from irreducible finite-dimensional Lorentz and SU(N) representations realized without spin and isospin indices. We use the lowest weight module structure of our su(2,2/N) representations to present a general procedure (adapted from the semisimple Lie algebra case) for the canonical construction of invariant differential operators closely related to the reducible (indecomposable) representations. All conformal supercovariant derivatives are obtained in this way. Examples of higher order invariant differential operators are given.     

Infinite dimensional groups and Riemann surface field theories

We show how to obtain positive energy representations of the groupG of smooth maps from a union of circles toU(N) from geometric data associated with a Riemann surface having these circles as boundary. Using covering spaces we can reduce to the case whereN=1. Then our main result shows that Mackey induction may be applied and yields representations of the connected component of the identity ofG which have the form of a Fock representation of an infinite dimensional Heisenberg group tensored with a finite dimensional representation of a subgroup isomorphic to the first cohomology group of the surface obtained by capping the boundary circles with discs. We give geometric sufficient conditions for the correlation functions to be positive definite and derive explicit formulae for them and for the vacuum (or cyclic) vector. (This gives a geometric construction of correlation functions which had been obtained earlier using tau functions.) By choosing particular functions inG with non-zero winding numbers on the boundary we obtain analogues of vertex operators described by Segal in the genus zero case. These special elements ofG (which have a simple interpretation in terms of function theory on theRiemann surface) approximate fermion (or Clifford algebra) operators. They enable a rigorous derivation of a form of boson-fermion correspondence in the sense that we construct generators of a Clifford algebra from the unitaries representing these elements ofG.     

Exact matrix elements for general two-body central-force interactions,expressed as sums of products

ABSTRACT

In a manner similar to but distinct from concurrent tensor efforts in electronic structure, it is shown that the Laplace transform can serve as a generator for a sum-of-products (SOP) form that allows one to write essentially any function of distance between two particles (i.e. any central force potential) as an exact two-body matrix. In particular, exact expressions for the Coulomb, Yukawa and long-range Ewald two-body operators are evaluated in a band-limited (Sinc function) basis. The resultant exact, full-basis, SOP representations for these interaction potentials – acting in conjunction with an external harmonic confining field – are validated via comparison with energy eigenstate solutions obtained via an independent calculation based on separation of variables. The new two-body matrix representations may have substantial impact in any of the many disciplines in which pair-wise central force interactions are relevant – especially, electronic structure and dynamics.     

Degenerations and Representations of Twisted Shibukawa–Ueno R-Operators

We study degenerations of the Belavin R-matrices via the infinite dimensional operators defined by Shibukawa–Ueno. We define a two-parameter family of generalizations of the Shibukawa–Ueno R-operators. These operators have finite dimensional representations which include Belavin's R-matrices in the elliptic case, a two-parameter family of twisted affinized Cremmer–Gervais R-matrices in the trigonometric case, and a two-parameter family of twisted (affinized) generalized Jordanian R-matrices in the rational case. We find finite dimensional representations which are compatible with the elliptic to trigonometric and rational degeneration. We further show that certain members of the elliptic family of operators have no finite dimensional representations. These R-operators unify and generalize earlier constructions of Felder and Pasquier, Ding and Hodges, and the authors, and illuminate the extent to which the Cremmer–Gervais R-matrices (and their rational forms) are degenerations of Belavin's R-matrix.     

The application of the QSSA via reaction lumping for the reduction of complex hydrocarbon oxidation mechanisms

The quasi-steady-state approximation (QSSA) has been widely applied for the purposes of chemical kinetic model reduction. Although it is essentially a low-order approximation, it can be shown to lead to significant reductions in the number of fast variables within a mechanism without significant loss of accuracy for model predictions. Due to the couplings between QSSA expressions, the species are commonly solved for using numerical inner iteration techniques. Therefore, although the stiffness of the model system can be reduced, there is a computational overhead in solving the often nonlinear QSSA equations. Greater computational savings can be made where QSS species can be removed from the chemical model via explicit analytical expressions. In many cases these expressions are equivalent to reaction lumping. Where such reaction lumping can be achieved, a reduced mechanism in standard kinetic form can be developed, which contains new lumped reaction rate coefficients, but leads to the removal of QSS species. This paper describes such an approach for mechanisms describing the oxidation of the hydrocarbon fuels n-heptane and cyclohexane, and shows that significant reductions in both species and reactions can be achieved, leading to substantial computational speed-ups. The resulting schemes clearly demonstrate the main atomic flux patterns within the oxidation process. Patterns related to the time-scales of hydrocarbon radical species within alkane oxidation mechanisms are discussed, as well as the potential significance of non-QSS radicals in determining ignition behaviour.     

Particle diffraction by a thin rigid crystal lattice: A formal approach

The diffraction process of a particle by a thin rigid crystal is considered. An integral equation is derived for the particle wave function φ which is quite suitable to obtain physical and mathematical properties. A class of potentials is presented for which the integral equation can be solved by means of the Fredholm theory. The convergence of the Born series for φ is studied, as well as the existence and convergence properties of the transmission and reflection amplitudes T^±. Results are given about φ and T^±: (i) at high energies, (ii) at those special energies such that new diffracted beams appear, and (iii) at glancing incidence on the crystal. Analyticity properties of T^± as functions of the energy are derived and analytic representations for them are presented. The diffraction process when the particle is being simultaneously accelerated by a uniform electric field is also considered. Finally, the generalization to the case of an imperfect thin crystal is treated.     

Infinite-dimensional representations of the Lsorentz group: The complete series

The method used by Carmeli to obtain a new form for the principal series of representations of the groupSL(2, C) is further generalized to all completely irreducible (finite and infinite-dimensional) representations of that group. This is done, following Naimark, by extending the meaning of one of the parameters appearing in the formula for the operators of the principal series of representations. As a result a new form for the complete series of representations of the groupSL(2, C) is obtained which describes the transformation law of an infinite set of quantities under the group translation in a way which is very similar, but as a generalization, to the way spinors appear in the finite-dimensional case. The finite-dimensional representation is then discussed in details and the relation between the new set of quantities (which becomes finite in this case) and 2-component spinors is found explicitly.     

Master Analytic Representations and unified representation theory of certain orthogonal and pseudo-orthogonal groups

The representation theory of the groupsSO(5),SO(4, 1),SO(6) andSO(5, 1) is studied using the method of Master Analytic Representations (MAR). It is shown that a single analytic expression for the matrix elements of the generators ofSO(n+1) andSO(n, 1) in anSO(n) basis yields all the unitary representations (forn=4,5); and that the compact and non-compact groups have essentially the same analytic representation. Once the MAR of a group is worked out, the search for the unitary irreducible representations is reduced to a purely arithmetic operation. The utmost care has been exercised to conduct the discussions at an elementary level: knowledge of simple angular momentum theory is the only prerequisite.Work supported in part by the National Science Foundation.Work supported in part by the U.S. Atomic Energy Commission.