Riemannian Geometry of Quantum Groups¶and Finite Groups with Nonuniversal Differentials

We construct noncommutative “Riemannian manifold” structures on dual quasitriangular Hopf algebras such as ℂ _q [SU ₂] with its standard bicovariant differential calculus, using the quantum frame bundle approach introduced previously. The metric is provided by the braided-Killing form on the braided-Lie algebra on the tangent space and the n-bein by the Maurer–Cartan form. We also apply the theory to finite sets and in particular to finite group function algebras ℂ[G] with differential calculi and Killing forms determined by a conjugacy class. The case of the permutation group ℂ[S ₃] is worked out in full detail and a unique torsion free and cotorsion free or “Levi–Civita” connection is obtained with noncommutative Ricci curvature essentially proportional to the metric (an Einstein space). We also construct Dirac operators in the metric background, including on finite groups such as S ₃. In the process we clarify the construction of connections from gauge fields with nonuniversal calculi on quantum principal bundles of tensor product form. Received: 22 June 2000 / Accepted: 26 August 2001     

Sum-of-Squares Results for Polynomials Related to the Bessis–Moussa–Villani Conjecture

We show that the polynomial S _m,k (A,B), that is the sum of all words in noncommuting variables A and B having length m and exactly k letters equal to B, is not equal to a sum of commutators and Hermitian squares in the algebra R〈X,Y〉, where X ²=A and Y ²=B, for all even values of m and k with 6≤k≤m−10, and also for (m,k)=(12,6). This leaves only the case (m,k)=(16,8) open. This topic is of interest in connection with the Lieb–Seiringer formulation of the Bessis–Moussa–Villani conjecture, which asks whether Tr (S _m,k (A,B))≥0 holds for all positive semidefinite matrices A and B. These results eliminate the possibility of using “descent + sum-of-squares” to prove the BMV conjecture.     

Noncommutative Manifolds from Graph and <Emphasis Type="Italic">k</Emphasis>-Graph <Emphasis Type="Italic">C</Emphasis>*-Algebras

In [PRen] we constructed smooth (1, ∞)-summable semifinite spectral triples for graph algebras with a faithful trace, and in [PRS] we constructed (k, ∞)-summable semifinite spectral triples for k-graph algebras. In this paper we identify classes of graphs and k-graphs which satisfy a version of Connes’ conditions for noncommutative manifolds.     

Static charged spheres with anisotropic pressure in general relativity

We report a generalization of our earlier formalism [Pramana, 54, 663 (1998)] to obtain exact solutions of Einstein-Maxwell’s equations for static spheres filled with a charged fluid having anisotropic pressure and of null conductivity. Defining new variables: w=(4π/3)(ρ+ε)r ², u=4πξr ², v _r=4πp _r r ², v _⊥=4πp _⊥ r ²[ρ, ξ(=−(1/2)F ₁₄ F ¹⁴), p _r, p _⊥ being respectively the energy densities of matter and electrostatic fields, radial and transverse fluid pressures whereas ε denotes the eigenvalue of the conformal Weyl tensor and interpreted as the energy density of the free gravitational field], we have recast Einstein’s field equations into a form easy to integrate. Since the system is underdetermined we make the following assumptions to solve the field equations (i) u=v _r=(a ²/2κ)r ⁿ⁺², v _⊥=k ₁ v _r, w=k ₂ v _r; a ², n(>0), k ₁, k ₂ being constants with κ=((k ₁+2)/3+k ₂) and (ii) w+u=(b ²/2)r ⁿ⁺², u=v _r, v _⊥−v _r=k, with b and k as constants. In both cases the field equations are integrated completely. The first solution is regular in the metric as well as physical variables for all values of n>0. Even though the second solution contains terms like k/r ² since Q(0)=0 it is argued that the pressure anisotropy, caused by the electric flux near the centre, can be made to vanish reducing it to the generalized Cooperstock-de la Cruz solution given in [14]. The interior solutions are shown to match with the exterior Reissner-Nordstrom solution over a fixed boundary. Dedicated to Prof. F A E Pirani.     

Automorphisms of the affineSU(3) fusion rules

We classify the automorphisms of the (chiral) level-k affineSU(3) fusion rules, for any value ofk, by looking for all permutations that commute with the modular matricesS andT. This can be done by using the arithmetic of the cyclotomic extensions where the problem is naturally posed. Whenk is divisible by 3, the automorphism group (Z ₂) is generated by the charge conjugationC. Ifk is not divisible by 3, the automorphism group (Z ₂×Z ₂) is generated byC and the Altschüler-Lacki-Zaugg automorphism. Although the combinatorial analysis can become more involved, the techniques used here forSU(3) can be applied to other algebras.     

A global version of the quantum duality principle

The “quantum duality principle” states that a quantisation of a Lie bialgebra provides also a quantisation of the dual formal Poisson group and, conversely, a quantisation of a formal Poisson group yields a quantisation of the dual Lie bialgebra as well. We extend this to a much more general result: namely, for any principal ideal domainR and for each primepεR we establish an “inner” Galois’ correspondence on the categoryHA of torsionless Hopf algebras overR, using two functors (fromHA to itself) such that the image of the first and the second is the full subcategory of those Hopf algebras which are commutative and cocommutative, modulop, respectively (i.e., they are“quantum function algebras” (=QFA) and“quantum universal enveloping algebras” (=QUEA), atp, respectively). In particular we provide a machine to get two quantum groups — a QFA and a QUEA — out of any Hopf algebraH over a fieldk: apply the functors tok[ν] ⊗k H forp=ν. A relevant example occurring in quantum electro-dynamics is studied in some detail. Presented at the 10th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 21–23 June 2001     

States as Morphisms

Using elementary categorical methods, we survey recent results concerning D-posets (equivalently effect algebras) of fuzzy sets and the corresponding category ID in which states are morphisms. First, we analyze the canonical structures carried by the unit interval I = [0,1] as the range of states and the impact of “states as morphisms” on the probability domains. Second, we analyze categories of various quantum and fuzzy structures and their relationships. Third, we describe some basic properties of ID and show that traditional probability domains such as fields of sets and bold algebras can be viewed as full subcategories of ID and probability measures on fields of sets and states on bold algebras become morphisms. Fourth, we discuss the categorical aspects of the transition from classical to fuzzy probability theory. We conclude with some remarks about generalized probability theory based on ID.     

Neighborhood preferences in random matching problems

We consider a class of random matching problems where the distance between two points has a probability law which, for a small distance l, goes like l^r. In the framework of the cavity method, in the limit of an infinite number of points, we derive equations for p_k, the probability for some given point to be matched to its kth nearest neighbor in the optimal configuration. These equations are solved in two limiting cases: r = 0 -- where we recover p _k = 1/2^k, as numerically conjectured by Houdayer et al. and recently rigorously proved by Aldous -- and r→ + ∞. For 0 < r < + ∞, we are not able to solve the equations analytically, but we compute the leading behavior of p_k for large k. Received 14 February 2001     

Black holes with metric and fields of the same form

We present two rotating black hole solutions with axion ξ, dilaton f{\phi} and two U(1) vector fields. Starting from a non-rotating metric with three arbitrary parameters, which we have found previously, and applying the “Newman–Janis complex coordinate trick” we get a rotating metric g _μν with four arbitrary parameters namely the mass M, the rotation parameter a and the charges electric Q _E and magnetic Q _M . Then we find a solution of the equations of motion having this g _μν as metric. Our solution is asymptotically flat and has angular momentum J = M a, gyromagnetic ratio g = 2, two horizons, the singularities of the solution of Kerr, axion and dilaton singular only when r = a cos θ = 0 etc. By applying to our solution the S-duality transformation we get a new solution, whose axion, dilaton and vector fields have one more parameter. The metrics, the vector fields and the quantity l = x+ie^-2f{\lambda=\xi+ie^{-2\phi}} of our solutions and the solution of: Sen for Q _E , Sen for Q _E and Q _M , Kerr–Newman for Q _E and Q _M , Kerr, Reference Kyriakopoulos [Class. Quantum Grav. 23:7591, 2006, Eqs. (54–57)], Shapere, Trivedi and Wilczek, Gibbons–Maeda–Garfinkle–Horowitz–Strominger, Reissner–Nordstr?m, Schwarzschild are the same function of a, and two functions ρ ² = r(r + b) + a ² cos² θ and Δ = r(r + b) − 2Mr + a ² + c, of a, b and two functions for each vector field, and of a, b and d respectively, where a, b, c and d are constants. From our solutions several known solutions can be obtained for certain values of their parameters. It is shown that our two solutions satisfy the weak the dominant and the strong energy conditions outside and on the outer horizon and that all solutions with a metric of our form, whose parameters satisfy some relations satisfy also these energy conditions outside and on the outer horizon. This happens to all solutions given in the “Appendix”. Mass formulae for our solutions and for all solutions which are mentioned in the paper are given. One mass formula for each solution is of Smarr’s type and another a differential mass formula. Many solutions with metric, vector fields and λ of the same functional form, which include most physically interesting and well known solutions, are listed in an “Appendix”.     

On the Quantum Invariants for the Spherical Seifert Manifolds

We study the Witten–Reshetikhin–Turaev SU(2) invariant for the Seifert manifolds S ³/Gamma where Γ is a finite subgroup of SU(2). We show that the WRT invariants can be written in terms of the Eichler integral of modular forms with half-integral weight, and we give an exact asymptotic expansion of the invariants by use of the nearly modular property of the Eichler integral. We further discuss that those modular forms have a direct connection with the polyhedral group by showing that the invariant polynomials of modular forms satisfy the polyhedral equations associated to Γ.     

Vortex Condensates¶for the SU(3) Chern–Simons Theory

We investigate SU(3)-periodic vortices in the self-dual Chern–Simons theory proposed by Dunne in [13, 15]. At the first admissible non-zero energy level E= 2 π, and for each (broken and unbroken) vacuum state φ₍₀₎ of the system, we find a family of periodic vortices asymptotically gauge equivalent to φ₍₀₎, as the Chern–Simons coupling parameter k→ 0. At higher energy levels, we show the existence of multiple gauge distinct periodic vortices with at least one of them asymptotically gauge equivalent to the (broken) principal embedding vacuum, when k→ 0. Received: 23 October 1999 / Accepted: 14 March 2000     

Vertex Operators – From a Toy Model to Lattice Algebras

Within the framework of the discrete Wess–Zumino–Novikov–Witten theory we analyze the structure of vertex operators on a lattice. In particular, the lattice analogues of operator product expansions and braid relations are discussed. As the main physical application, a rigorous construction for the discrete counterpart g _n $ of the group valued field g(x) is provided. We study several automorphisms of the lattice algebras including discretizations of the evolution in the WZNW model. Our analysis is based on the theory of modular Hopf algebras and its formulation in terms of universal elements. Algebras of vertex operators and their structure constants are obtained for the deformed universal enveloping algebras . Throughout the whole paper, the abelian WZNW model is used as a simple example to illustrate the steps of our construction. Received: 16 December 1996 / Accepted: 5 May 1997     

An affine extension of non-crystallographic Coxeter groups with applications in the theory of quasicrystals and integrable systems

Similarly as in the theory of Kac-Moody algebras, affine extensions of the non-crystallographic Coxeter groupsH _k, (k=2, …, 4) can be derived via an appropriate extension of the Cartan matrix. These groups lead to novel applications in the theory of quasicrystals and integrable models. In the former case, a new model for quasicrystals with five-fold symmetries could be established; in the latter case, subgroups have been used to obtain a Calogero model related to a non-integrally laced group. Presented at the DI-CRM Workshop held in Prague, 18–21 June 2000. Financial support through a European-Union Marie Curie fellowship is gratefully acknowledged.     

Fusion rings and geometry

The algebraic structure of fusion rings in rational conformal field theories is analyzed in detail in this paper. A formalism which closely parallels classical tools in the study of the cohomology of homogeneous spaces is developed for fusion rings, in general, and for current algebra theories, in particular. It is shown that fusion rings lead to a natural orthogonal polynomial structure. The rings are expressed through generators and relations. The relations are then derived from some potentials leading to an identification of the fusion rings with deformations of affine varieties. In general, the fusion algebras are mapped to affine varieties which are the locus of the relations. The connection with modular transformations is investigated in this picture. It is explained how chiral algebras, arising inN=2 superconformal field theory, can be derived from fusion rings. In particular, it is argued that theories of the typeSU(N) _k /SU(n–1) are theN=2 counterparts of Grassmann manifolds and that there is a natural identification of the chiral fields with Schubert varieties, which is a graded algebra isomorphism.Supported in part by NSF grant PHY 89-04035 supplemented by funds from NASA     

Perturbation approach for equation of state for hard-sphere and Lennard–Jones pure fluids

In this paper we have established the equation of state (EOS) for liquids. The EOS was established for hard-sphere (HS) fluid along with Lennard–Jones (LJ) fluid incorporating perturbation techniques. The calculations are based on suitable axiomatic functional forms for surface tension S _m (r), r ≥ d/2 with intermolecular separation r, as a variable, and m is an arbitrary real number (pole). The results for βP/ρ from the present EOS thus obtained are compared with Percus-Yevick (PY), scaled particle theory (SPT), and Carnahan–Starling (CS). In addition, we have found a simple EOS for the HS fluid in the region which represents the simulation data accurately.     

Effect of an edge field on the acceptance of a quadrupole mass filter operating at the bottom vertex of the stability rectangle

A quadrupole mass filter (QMF) can operate with a large acceptance and high transmission at the bottom vertex S (a=2.5210, q=2.8153) of the stability quadrilateral. The combined acceptance at a level of 50% transmission and a resolving power of 100 equals 2.0×10⁻³ r ₀ ⁴ f ², which is comparable to the acceptance (5.1×10⁻³ r ₀ ⁴ f ²) of the standard operating regime of a QMF in the first stability region under the same computational conditions and optimal on-axis ion velocity. The acceptance is approximately three times higher in the presence of edge fields than in their absence. The optimal on-axis ion energy equals 1.15r ₀ f, where r ₀ is the radius of the field (the radius of the inscribed circle between the vertices of the electrodes) and f is the working frequency. In the gas-analysis regime a sensitivity of 10⁻⁵ A/Pa is achieved on a mass filter with rod length and rod diameter of 15 cm and 8 mm, respectively, frequency f=1 MHz, and field radius r ₀=0.35 cm. Zh. Tekh. Fiz. 67, 121–124 (October 1997)     

Dynamics and sedimentation velocity in charge-stabilized colloidal suspensions

Summary Charge-stabilized suspensions are characterized by the strong electrostatic interactions between the particles so that rather dilute systems may exhibit strong correlation resulting in a well-developed short-range order. This microstructure, quantitatively described by the pair distribution functiong(r), is rather different from that of (uncharged) hard spheres. It is shown how this difference affects the ?hydrodynamic function?H(k), which appears in the expression for the first cumulant Γ(k)=k ² D _eff(k)=k ² H(k)/S(k) of the dynamic autocorrelation function. Without hydrodynamic interaction,H(k)=D ⁰, which is the free-diffusion coefficient. Using pairwise additive hydrodynamic interaction and the lowest-order many-body theory of hydrodynamic interaction, it is found thatH(k) can deviate considerably fromD ⁰ even for systems of volume fractions ϕ as low as 10⁻³. These effects are more pronounced for collective diffusion than for self-diffusion. SinceH(k=0) is closely related to the sedimentation velocity, we have studied this quantity as a function of volume fraction. It is found that (H(0)/D ⁰) −1 scales asφ ^1/3 at low ϕ in salt-free suspensions. Paper presented at the I International Conference on Scaling Concepts and Complex Fluids, Copanello, Italy, July 4–8, 1994.     

On α-Induction, Chiral Generators and¶Modular Invariants for Subfactors

We consider a type III subfactor N⊂N of finite index with a finite system of braided N-N morphisms which includes the irreducible constituents of the dual canonical endomorphism. We apply α-induction and, developing further some ideas of Ocneanu, we define chiral generators for the double triangle algebra. Using a new concept of intertwining braiding fusion relations, we show that the chiral generators can be naturally identified with the α-induced sectors. A matrix Z is defined and shown to commute with the S- and T-matrices arising from the braiding. If the braiding is non-degenerate, then Z is a “modular invariant mass matrix” in the usual sense of conformal field theory. We show that in that case the fusion rule algebra of the dual system of M-M morphisms is generated by the images of both kinds of α-induction, and that the structural information about its irreducible representations is encoded in the mass matrix Z. Our analysis sheds further light on the connection between (the classifications of) modular invariants and subfactors, and we will construct and analyze modular invariants from SU(n) _k loop group subfactors in a forthcoming publication, including the treatment of all SU(2) _k modular invariants. Received: 13 April 1999 / Accepted: 13 July 1999     

Real forms of extended Kac–Moody symmetries and higher spin gauge theories

We consider the relation between higher spin gauge fields and real Kac–Moody Lie algebras. These algebras are obtained by double and triple extensions of real forms \mathfrakg₀{\mathfrak{g}_0} of the finite-dimensional simple algebras \mathfrakg{\mathfrak{g}} arising in dimensional reductions of gravity and supergravity theories. Besides providing an exhaustive list of all such algebras, together with their associated involutions and restricted root diagrams, we are able to prove general properties of their spectrum of generators with respect to a decomposition of the triple extension of \mathfrakg₀{\mathfrak{g}_0} under its gravity subalgebra \mathfrakgl(D,\mathbb R){\mathfrak{gl}(D,\mathbb {R})} . These results are then combined with known consistent models of higher spin gauge theory to prove that all but finitely many generators correspond to non-propagating fields and there are no higher spin fields contained in the Kac–Moody algebra.     

Modular Classes of Regular Twisted Poisson Structures on Lie Algebroids

We derive a formula for the modular class of a Lie algebroid with a regular twisted Poisson structure in terms of a canonical Lie algebroid representation of the image of the Poisson map. We use this formula to compute the modular classes of Lie algebras with a twisted triangular r-matrix. The special case of r-matrices associated to Frobenius Lie algebras is also studied.