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     

Twist deformations for algebras of motion

We study the twist deformations of algebras of motiong ^H ⊂ sl(N) with the Cartan subalgebraH(g^H) equal toH(sl(N)). The proposed deformations are maximal in the sense that their carrier algebrasg _c coincide withg ^H. The algebraic properties are demonstrated forg ^H ⊂ sl(5). Presented at the 11th Colloquium “Quantum Groups and Integrable Systems”, Prague, 20–22 June 2002. This work has been partially supported by the Russian Foundation for Fundamental Research under the grant N 00-01-00500.     

Full chains of twists for orthogonal algebras

We show that for some Hopf subalgebras inU _F(so(M)) nontrivially deformed by a twistF it is possible to find the nonlinear primitive copies. This enlarges the possibilities to construct chains of twists. For orthogonal algebraU(so(M)) we present a method to compose the full chains with carrier space as large as the Borel subalgebraB(so(M)). These chains can be used to construct the new deformed Yangians. Presented at the 9-th Colloquium “Quantum Groups and Integrable Systems”, Prague, 22–24 June 2000. This work was partially supported by the Russian Foundation for Basic Research under the grant 00-01-00500 (VDL) and 98-01-00310 (PPK).     

Deformed yangians and integrable models

Twisted Hopf algebraslξ(2) gives rise to a deformation of the Yangiany(sl(2)). The corresponding deformations of the integrableXXX-spin chain and the Gaudin model are discussed. Presented at 6th International Colloquium on Quantum Groups: “Quantum Groups and Integrables Systems”, Prague, 19–21 June 1997. This research was supported by the Swedish National Research Council and RFFI grant 96-01-00851.     

Bicrossproduct structure and graded contractions of deformed algebras

It is shown that all members in the family of deformed Hopf algebras corresponding to the graded contractions of the inhomogeneous algebrasiso(p,q),p +q=N, have a bicrossproduct structure. Presented at the 6th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 19–21 June 1997. This research has been partially supported by a research grant from the Spanish CICYT.     

Tensor operators for quantum algebras

Tensor operators are discussed for Hopf algebras and, in particular, for a quantum (q-deformed) algebraUq(g), whereg is any simple finite-dimensional or affine Lie algebra. These operators are defined via an adjoint action in a Hopf algebra. There are two types of the tensor operators which correspond to two coproducts in the Hopf algebra. In the case of tensor products of two tensor operators one can obtain 8 types of the tensor operators and so on. We prove the relations which can be a basis for a proof of the Wigner-Eckart theorem for the Hopf algebras. It is also shown that in the case ofUq(g) a scalar operator can be differed from an invariant operator but atq=1 these operators coincide. Presented at the 10th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 21–23 June 2001. Supported by Russian Foundation for Fundamental Research, grant 99-01-01163, and by INTAS-00-00055.     

On the integral construction of the universal R-matrix for quantum current superalgebraUq (oŝp(1,2))

We discuss the problems in the construction of the universal R-matrix for the the Drinfeld current realization of quantum affine superalgebrasUq (oŝp(1,2)), where we try to present the universal R-matrix for the corresponding “Drinfeld” comultiplication in the form of certain integrals over current operators with specially chosen contours. Presented at the 10th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 21–23 June 2001. Author’s research is partially supported by the Summer Research Fellowship and the Taft Foundation at the University of Cincinnati.     

Vafa-Witten Estimates for Compact Symmetric Spaces

We give an optimal upper bound for the first eigenvalue of the untwisted Dirac operator on a compact symmetric space G/H with rk G - rk H ≤ 1 with respect to arbitrary Riemannian metrics. We also prove a rigidity statement. Supported in part by DFG special programme “Global Differential Geometry”     

C*-Algebras associated with one dimensional almost periodic tilings

For each irrational number, 0<α<1, we consider the space of one dimensional almost periodic tilings obtained by the projection method using a line of slope α. On this space we put the relation generated by translation and the identification of the “singular pairs”. We represent this as a topological spaceX _α with an equivalence relationR _α. OnR _α there is a natural locally Hausdorff topology from which we obtain a topological groupoid with a Haar system. We then construct the C^*-algebra of this groupoid and show that it is the irrational rotation C^*-algebra,A _α. Research supported by the Natural Sciences and Engineering Research Council of Canada and the Fields Institute for Research in Mathematical Sciences.     

Reflection equation- and FRT-type algebras

We introduce two types of algebras which include respectively the well known reflection equation (RE) and Faddeev-Reshetikhin-Takhtayan algebras associated with a quasitriangular Hopf algebraH. We show that these two types of algebras are twist-equivalent. It follows that a RE algebra is a module algebra over a twisted tensor square ofH. We present some applications to the equivariant quantization. Presented at the 11th Colloquium “Quantum Groups and Integrable Systems”, Prague, 20–22 June 2002.     

Cohomologies and peripheric chains

The deformation properties of Lie subalgebrasLεg that are carriers for twistsF : U(g)→UF(g) are tightly connected with the possibility to apply additional twisting deformationsFR andFJ┴ toUF(g) and quantize new sets ofγ-matrices. Presented at the 10th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 21–23 June 2001. This work has been partially supported by the Russian Foundation for Fundamental Research under the grant N 00-01-00500.     

Kauffman bracket of plane curves

We lower the Kauffman bracket for links in a solid torus (see [16]) to generic plane fronts. It turns out that the bracket can be entirely defined in terms of a front itself without using the Legendrian lifting. We show that all the coefficients of the lowered bracket are in fact Vassilev type invariants of Arnold'sJ ⁺-theory [3, 4]. We calculate their weight systems. As a corollary we obtain that the first coefficient is essentially the quantum deformation of the Bennequin invariant introduced recently by M. Polyak [19]. Partially supported by the Fellowship “Emma e Giovanni Sansone,” the International Science Foundation and the Russian Foundation for Fundamental Research. Supported by an RDF grant of The University of Liverpool.     

Super Universality of the Quantum Hall Effect and the?“Large N Picture” of the ? Angle

It is shown that the “massless chiral edge excitations” are an integral and universal aspect of the low energy dynamics of the ϑ vacuum that has historically gone unnoticed. Within the SU(M+N)/S(U(M)×U(N)) non-linear sigma model we introduce an effective theory of “edge excitations” that fundamentally explains the quantum Hall effect. In sharp contrast to the common beliefs in the field our results indicate that this macroscopic quantization phenomenon is, in fact, a super universal strong coupling feature of the ϑ angle with the replica limit M=N=0 only playing a role of secondary importance. To demonstrate super universality we revisit the large N expansion of the CP ^N−1 model. We obtain, for the first time, explicit scaling results for the quantum Hall effect including quantum criticality of the quantum Hall plateau transition. Consequently a scaling diagram is obtained describing the cross-over between the weak coupling “instanton phase” and the strong coupling “quantum Hall phase” of the large N theory. Our results are in accordance with the “instanton picture” of the ϑ angle but fundamentally invalidate all the ideas, expectations and conjectures that are based on the historical “large N picture.”     

Quantization of triangular Lie bialgebras

We derive a universal twisting element for an arbitrary triangularγ-matrix using a simple analogue of the Fedosov quantization method. Presented at the 11th Colloquium “Quantum Groups and Integrable Systems”, Prague, 20–22 June 2002. The work of SLL is partially supported by RFBR grant 00-02-17-956 and the grant INTAS 00-262. The work of AASh is supported by RFBR grant 02-02-06879 and Russian Ministry of Education under the grant E-00-33-184. The work of VAD is partially supported by the grant INTAS 00-561 and by the Grant for Support of Scientific Schools 00-15-96557. The work of API is partially supported by the RFBR grant 00-01-00299.     

Deligne Products of Line Bundles over Moduli Spaces of Curves

We study Deligne products for forgetful maps between moduli spaces of marked curves by offering a closed formula for tautological line bundles associated to marked points. In particular, we show that the Deligne products for line bundles on the total spaces corresponding to “forgotten” marked points are positive integral multiples of the Weil-Petersson bundles on the base moduli spaces. Partially supported by the Japan Society for the Promotion of Science.     

Regular solutions of quantum Yang-Baxter equation from weak hopf algebras

Generalization of Hopf algebraslq (2) by weakening the invertibility of the generatorK, i.e., exchanging its invertibilityKK ⁻¹=1 to the regularity K K=K is studied. Two weak Hopf algebras are introduced: a weak Hopf algebrawslq (2) and aJ-weak Hopf algebravslq (2) which are investigated in detail. The monoids of group-like elements ofwslq (2) andvslq (2) are regular monoids, which supports the general conjucture on the connection betweek weak Hopf algebras and regular monoids. A quasi-braided weak Hopf algebraŪqw is constructed fromwslq (2). It is shown that the corresponding quasi-R-matrix is regular R^{w w}R^w=R^w. Presented at the 10th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 21–23 June 2001 Project (No. 19971074) supported by the National Natural Science Foundation of China.     

Hamilton-Jacobi formulation of the Harada gauged Floreanini-Jackiw action

We study the front-form Harada gauged Floreanini-Jackiw action and its BRST-anti-BRST symmetry within Hamilton-Jacobi formalism. Presented at the 11th Colloquium “Quantum Groups and Integrable Systems”, Prague, 20–22 June 2002. This paper is partially supported by the Scientific and Technical Research Council of Turkey.     

Majorana neutrino masses in the three-flavor Pauli model

A special Majorana model for three neutrino flavors is developed on the basis of the Pauli transformation group. In this model, the neutrinos possess a partially conserved generalized lepton (Pauli) charge that makes it possible to discriminate between neutrinos of different type. It is shown that, within the model in question, a transition from the basic “mass” representation, where the average value of this charge is zero, to the representation associated with physical neutrinos characterized by specific Pauli “flavor” charges establishes a relation between the neutrino mixing angles θ _{mix, 12}, θ _{mix, 23}, and θ _{mix, 13} and an additional relation between the Majorana neutrino masses. The Lagrangian mass part, which includes a term invariant under Pauli transformations and a representation-dependent term, concurrently assumes a “quasi-Dirac” form. With allowance for these relations, the existing set of experimental data on the features of neutrino oscillations makes it possible to obtain quantitative estimates for the absolute values of the neutrino masses and the 2β-decay mass parameter m _ββ and a number of additional constraints on the neutrino mixing angles.     

The interplay between “low” and “high” energy CP-violation in leptogenesis

We analyse, within the “flavoured” leptogenesis scenario of baryon asymmetry generation, the interplay of “low energy” CP-violation, originating from the PMNS neutrino mixing matrix U, and “high energy” CP-violation, which can be present in the matrix of neutrino Yukawa couplings, λ, and can manifest itself only in “high” energy scale processes. The type I see-saw model with three heavy right-handed Majorana neutrinos having a hierarchical spectrum is considered. The “orthogonal” parameterisation of the matrix of neutrino Yukawa couplings, which involves a complex orthogonal matrix R, is employed. In this approach the matrix R is the source of “high energy” CP-violation. Results for normal hierarchical (NH) and inverted hierarchical (IH) light neutrino mass spectrum are derived in the case of decoupling of the heaviest right-handed Majorana neutrino. It is shown that taking into account the contribution to Y _B due to the CP-violating phases in the neutrino mixing matrix U can change drastically the predictions for Y _B , obtained assuming that only “high energy” CP-violation from the R-matrix is operative in leptogenesis. In the case of the IH spectrum, in particular, there exist significant regions in the corresponding parameter space where the purely “high energy” contribution in Y _B plays a subdominant role in the production of baryon asymmetry compatible with the observations. Also at Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 1784 Sofia, Bulgaria.     

Yangians, quantum groups and solutions of the quantum dynamical Yang-Baxter equation

We construct Drinfel’d twists that define deformed Hopf structures. In particular, we obtain deformed double Yangians and dynamical double Yangians. Presented by D. Arnaudon at the 10th Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 21–23 June, 2001.