SU q (2) covariant\hat R-matrices for reducible representations

We consider SU _q (2) covariant -matrices for the reducible3 1 representation. There are three solutions to the Yang-Baxter equation. They coincide with the previously known -matrices for SO _q (3) and SO _q (3, 1). Also, they are the three -matrices which can be constructed by using four different SU _q (2) doublets. Only two of the three -matrices allow a differential structure on the reducible four-dimensional quantum space.     

Some Tensor Product Representations of Uq(s $${\hat l}{\kern 1pt} $$ 2) at Roots of Unity

When q is a root of unity, a triangular decomposition of U _q(s ₂) is given and irreducibility conditions concerning some tensor product representations of U _q(s ₂) are presented. Their connection with physics is also pointed out.     

Coordinates of the quantum plane as q-tensor operators in (\mathfrak{s}\mathfrak{u}(2)*\mathfrak{s}\mathfrak{u}(2))

The relation between the set of transformations of the quantum plane and the quantum universal enveloping algebra U _q (u(2)) is investigated by constructing representations of the factor algebra U _q (u(2))* . The noncommuting coordinates of , on which U _q (2) * U _q (2) acts, are realized as q-spinors with respect to each U _q (u(2)) algebra. The representation matrices of U _q (2) are constructed as polynomials in these spinor components. This construction allows a derivation of the commutation relations of the noncommuting coordinates of directly from properties of U _q (u(2)). The generalization of these results to U _q (u(n)) and is also discussed.     

A classification of the unitary irreducible representations ofSU(N, 1)

All inequivalent continuous unitary irreducible representations ofS U(N, 1) (N2) have been determined and classified. The matrix elements of the infinitesimal generators realized on a certain Hilbert space have been derived. Representations of the groups ,S U(N, 1)/Z _N+1, andU(N, 1) are classified in a similar manner.     

Four-quark states and nucleon-antinucleon annihilation within the quark model with QCD vacuum-induced interaction

The spectrum of , J^p=0⁺, 2⁺ mesons is discussed. We have shown that due to instanton-induced forces the physical states are strong mixtures of theSU _f (3) group basis states. The cross-sections for annihilation of the system into mesons are obtained. Thea ₀(980) meson is considered as meson consisting of 9 _f and 36 _f plets. The branchings are also predicted for the cross-sections for production of thea ₀(980) and tensor mesons in annihilation.     

From the deuteron to deusons,an analysis of deuteronlike meson-meson bound states

A systematic study of possible deuteronlike twomeson bound states,deusons, is presented. Previous arguments that many such bound states may exist are elaborated with detailed arguments and numerical calculations including, in particular, the tensor potential. This tensor potential which is crucial for the deuteron binding is shown to be very important also in the mesonic case. Especially, in the pseudoscalar³ P ₀ pseudoscalar-vector and vector-vector channels the important observation is made that the centrifugal barrier from theP-wave can be overcome by the 1/r ² and 1/r ³ terms of the tensor potential. In the heavy meson sector one-pion exchange alone is strong enough to form at least deuteronlike and composites bound by approximately 50 MeV. Composites of and states bound by pion exchange alone are expected near the thresholds, while in the light meson sector one generally needs some additional short range attraction to form bound states. The quantum numbers of these states areI=0, andJ ^PC=0^–+, 1⁺⁺ for the states andI=0,J ^PC=0⁺⁺, O^–+, 1^+– and 2⁺⁺ for the composites. In the states: ^b (10545), _b1(10562) are predicted and in , one finds the states: ^b (10590), _bQ (10582),h _b(10608), _b2(10602). Near the threshold the states: ^c (3870), _c0(3870) are predicted, and near the threshold one finds the states: _b0(4015), ^c (4015),h _c(4015), _c2(4015). Within the light meson sector pion exchange gives strong attraction for and systems with quantum numbers where the best non- candidates exist, although pion exchange alone is not strong enough to support such bound states. Thus, although one cannot conclude with certainty it to be the case, this fact does favour the picture that the (1440) and thef ₁ (1420) are mainly composites and thef ₀(1710) mainly a bound state, while thef ₀(1515) andf ₂(1520) could be predominantly composites. If the predicted and states are found, these would support this interpretation of the light states. In channels with exotic flavour orCP quantum numbers pion exchange is generally repulsive or quite weak. Therefore one does not expect that such deuteronlike bound states exist, althoughB*B* may be an exception.     

Minimal affinizations of representations of quantum groups: the irregular case

Let be a finite-dimensional complex simple Lie algebra and U_q( ) the associated quantum group (q is a nonzero complex number which we assume is transcendental). IfV is a finitedimensional irreducible representation of U_q( ), an affinization ofV is an irreducible representationVV of the quantum affine algebra U_q( ) which containsV with multiplicity one and is such that all other irreducible U_q( )-components ofV have highest weight strictly smaller than the highest weight ofV. There is a natural partial order on the set of U_q( ) classes of affinizations, and we look for the minimal one(s). In earlier papers, we showed that (i) if is of typeA, B, C, F orG, the minimal affinization is unique up to U_q( )-isomorphism; (ii) if is of typeD orE and is not orthogonal to the triple node of the Dynkin diagram of , there are either one or three minimal affinizations (depending on ). In this paper, we show, in contrast to the regular case, that if U_q( ) is of typeD ₄ and is orthogonal to the triple node, the number of minimal affinizations has no upper bound independent of .As a by-product of our methods, we disprove a conjecture according to which, if is of typeA _n,every affinization is isomorphic to a tensor product of representations of U_q( ) which are irreducible under U_q( ) (in an earlier paper, we proved this conjecture whenn=1).Both authors were partially supported by the NSF, DMS-9207701.     

f(R) Cosmology in the First Order Formalism

In the present work we consider those theories that are obtained from a Lagrangian density _T (R) = f(R){-g} + _M , that depends on the curvature scalar and a matter Lagrangian that does not depend on the connection, and apply Palatini's method to obtain the field equations. We start with a brief discussion of the field equations of the theory and apply them to a cosmological model described by the FRW metric. Then, we introduce an auxiliary metric to put the resultant equations into the form of GR with cosmological constant and coupling constant that are curvature depending. We show that we reproduce known results for the quadratic case. We find relations among the present values of the cosmological parameters q ₀, H ₀, and . Next we use a simple perturbation scheme to find the departure in angular diameter distance with respect to General Relativity. Finally, we use the observational data to estimate the order of magnitude of what is essentially the departure of f(R) from linearity. The bound that we find for f (0) is so huge that permit almost any f(R). This is in the nature of things: the effect of higher order terms in f(R) are strongly suppressed by power of Planck's time 8G ₀. In order to improve these bounds more research on mathematical aspects of these theories and experimental consequences is necessary.     

Demixing in Isotropic Binary Mixtures of Rodlike Macromolecules

A binary mixture of long rigid rods of diameters D _i and lengths L _i (i=1, 2) may demix into two isotropic phases, and we give necessary conditions on the molecular size parameters for this transition to exist. These conditions imply that the two diameters must be sufficiently unequal, D ₂/D ₁>( + )², or D ₂/D ₁<( + )², while the length ratio is limited to an interval f _–(D ₂/D ₁)<L ₂/L ₁<f ₊(D ₂/D ₁). The functions f _± are given explicitly.     

Understanding the scalar mesonq\bar q nonet

It is shown that one can fit the available data on thea _o(980),f _o(1300) andK _o ^* (1430) mesons as a distorted 0⁺⁺ nonet using very few (5–6) parameters and an improved version of the unitarized quark model. This includes all light two-pseudoscalar thresholds, constraints from Adler zeroes, flavour symmetric couplings, unitarity and physically acceptable analyticity. The parameters include a bare or mass, an over-all coupling constant, a cutoff and a strange quark mass of 100 MeV, which is in accord with expectations from the quark model. It is found that in particular for thea ₀(980) andf ₀(980) the component, in the wave function is large, i.e., for a large fraction of the time the state is transformed into a virtual pair. This component, together with a similar component of for thea ₀(980), and , and components for thef ₀(980), causes the substantial shift to a lower mass than what is naively expected from the component alone. Mass, width and mixing parameters, including sheet and pole positions, of the four resonances are given, with a detailed pedagogical discussion of their meaning.     

Higher spin BRS cohomology of supersymmetric chiral matter inD=4

We examine the BRS cohomology of chiral matter inN=1,D=4 supersymmetry to determine a general form of composite superfield operators which can suffer from supersymmetry anomalies. Composite superfield operators _{(a, b)} are products of the elementary chiral superfieldsS and and the derivative operatorsD , and . Such superfields _{(a, b)} can be chosen to have a symmetrized undotted indices _i and b symmetrized dotted indices . The result derived here is that each composite superfield _(a,b) is subject to potential supersymmetry anomalies ifa–b is an odd number, which means that _(a,b) is a fermionic superfield.     

Remarks on the Schur—Howe—Sergeev Duality

We establish a new Howe duality between a pair of two queer Lie superalgebras (q(m),q(n)). This gives a representation theoretic interpretation of a well-known combinatorial identity for Schur Q-functions. We further establish the equivalence between this new Howe duality and the Schur–Sergeev duality between q(n) and a central extension of the hyperoctahedral group H _k. We show that the zero-weight space of a q(n)-module with highest weight given by a strict partition of n is an irreducible module over the finite group parameterized by . We also discuss some consequences of this Howe duality.     

E(transverse optical)-mode properties in the LiTaO3: Nd crystal at room temperature

E(TO)-mode properties in LiTaO₃:Nd crystal were examined by analyzing the Raman spectra measured.E(TO) modes appear in the transverseA ₁ spectrum. Their intensities obviously increase in theE andE+A ₁ mixed-symmetry spectra but decrease in theE spectrum which shows new vibrational modes. In particular, in the transverse-E spectrum of y(xz) geometry, the properties ofE(TO) modes are similar to those of pure LiTaO₃ of the same geometry, whereas in the transverse-E spectrum in x(yz) geometry these modes are turned intoA ₁ (TO) modes. We attribute these properties to both the surface strain produced by mechanical polishing of the sample and the microstructural change of the LiTaO₃ crystal resulting from Nd doping.     

Existence of Resonances for Metric Scattering in Even Dimensions

Suppose _g is the (negative) Laplace–Beltrami operator of a Riemannian metric g on ⁿ which is Euclidean outside some compact set. It is known that the resolvent R()=(– _g –²)^–1, as the operator from L ² _comp( ⁿ ) to H ² _loc( ⁿ ), has a meromorphic extension from the lower half plane to the complex plane or the logarithmic plane when n is odd or even, respectively. Resonances are defined to be the poles of this meromorphic extension. We prove that when n is 4 or 6, there always exist infinitely many resonances provided that g is not flat. When n is greater than 6 and even, we prove the same result under the condition that the metric is conformally Euclidean or is close to the Euclidean metric.     

Quantum-Logics-Valued Measure Convergence Theorem

In this paper, the following quantum-logic valued measure convergence theorem is proved: Let (L ₁, 0, 1) be a Boolean algebra, (L ₂, , , 0, 1) be a quantum logic and { _n : n N} be a sequence of s-bounded (L ₂, , , 0, 1)-valued measures which are defined on (L ₁, 0, 1). If for each a (L ₁, 0, 1), { _n (a)} _{n N} is an order topology Cauchy sequence, when {v(a)} convergent to 0, { _n (a)} is order topology convergent to 0 for each n N, where v is a nonnegative finite additive measure which is defined on (L ₁, 0, 1), then when {v(a)} convergent to 0, { _n (a)} are order topology convergent to 0 uniformly with respect to n N.     

Generalization of the $$\mathcal{U} $$ q (gl(N)) Algebra and Staggered Models

We develop a technique for the construction of integrable models with a ₂ grading of both the auxiliary (chain) and quantum (time) spaces. These models have a staggered disposition of the anisotropy parameter. The corresponding Yang–Baxter equations are written down and their solution for the gl(N) case is found. We analyze in details the N = 2 case and find the corresponding quantum group behind this solution. It can be regarded as the quantum group , with a matrix deformation parameter q such that (q )² = q ². The symmetry behind these models can also be interpreted as the tensor product of the (–1)-Weyl algebra by an extension of _q (gl(N)) with a Cartan generator related to deformation parameter –1.     

Commutator Representations of Covariant Differential Calculi on Quantum Groups

Let (, d) be a first-order differential *-calculus on a *-algebra . We say that a pair (, F) of a *-representation of on a dense domain of a Hilbert space and a symmetric operator F on gives a commutator representation of if there exists a linear mapping : L( ) such that (adb) = (a)i[F, (b) ], a, b . Among others, it is shown that each left-covariant *-calculus of a compact quantum group Hopf *-algebra has a faithful commutator representation. For a class of bicovariant *-calculi on , there is a commutator representation such that F is the image of a central element of the quantum tangent space. If is the Hopf *-algebra of the compact form of one of the quantum groups SL _q (n+1), O _q (n), Sp _q (2n) with real trancendental q, then this commutator representation is faithful.     

Letter: On the Newtonian Limit in Gravity Models with Inverse Powers of R

I reconsider the problem of the Newtonian limit in nonlinear gravity models in the light of recently proposed models with inverse powers of R. Expansion around a maximally symmetric local background with curvature scalar R ₀ > 0 gives the correct Newtonian limit on length scales R ₀ ^–1/2 if the gravitational Lagrangian satisfies f(R ₀)f(R₀) 1, and I propose two models with f(R ₀) = 0.     

New quantum groups by double-bosonisation

We obtain new family of quasitriangular Hopf algebras via the author's recent double-bosonisation construction for new quantum groups. They are versions of U _q(su _n+1) with a fermionic rather than bosonic quantum plane of roots adjoined to U _q(su _n). We give the n = 2 case in detail. We also consider the anyonic-double of an anyonic ( ) braided group and the double-bosonisation of the free braided group in n variables.     

Exact Large Deviation Functional of a Stationary Open Driven Diffusive System: The Asymmetric Exclusion Process

We consider the asymmetric exclusion process (ASEP) in one dimension on sites i=1,...,N, in contact at sites i=1 and i=N with infinite particle reservoirs at densities _a and _b . As _a and _b are varied, the typical macroscopic steady state density profile ¯(x), x[a,b], obtained in the limit N=L(b–a), exhibits shocks and phase transitions. Here we derive an exact asymptotic expression for the probability of observing an arbitrary macroscopic profile , so that is the large deviation functional, a quantity similar to the free energy of equilibrium systems. We find, as in the symmetric, purely diffusive case q=1 (treated in an earlier work), that is in general a non-local functional of (x). Unlike the symmetric case, however, the asymmetric case exhibits ranges of the parameters for which is not convex and others for which has discontinuities in its second derivatives at (x)=¯(x). In the latter ranges the fluctuations of order in the density profile near ¯(x) are then non-Gaussian and cannot be calculated from the large deviation function.