Non-rationality of some fibrations associated to Klein surfaces

We study the polynomial fibration induced by the equation of the Klein surfaces obtained as quotient of finite linear groups of automorphisms of the plane; this surfaces are of type A, D, E, corresponding to their singularities. The generic fibre of the polynomial fibration is a surface defined over the function field of the line. We proved that it is not rational in cases D, E, although it is obviously rational in the case A. The group of automorphisms of the Klein surfaces is also described, and is linear and of finite dimension in cases D, E; this result being obviously false in case A.Comment: 18 page

Reconstruction of universal Drinfeld twists from representations

Universal Drinfeld twists are inner automorphisms which relate the coproduct of a quantum enveloping algebra to the coproduct of the undeformed enveloping algebra. Even though they govern the deformation theory of classical symmetries and have appeared in numerous applications, no twist for a semi-simple quantum enveloping algebra has ever been computed. It is argued that universal twists can be reconstructed from their well known representations. A method to reconstruct an arbitrary element of the enveloping algebra from its irreducible representations is developed. For the twist this yields an algebra valued generating function to all orders in the deformation parameter, expressed by a combination of basic and ordinary hypergeometric functions. An explicit expression for the universal twist of su(2) is given up to third order.Comment: 24 page

The anticommutator spin algebra, its representations and quantum group invariance

We define a 3-generator algebra obtained by replacing the commutators by anticommutators in the defining relations of the angular momentum algebra. We show that integer spin representations are in one to one correspondence with those of the angular momentum algebra. The half-integer spin representations, on the other hand, split into two representations of dimension j + 1/2. The anticommutator spin algebra is invariant under the action of the quantum group SO_q(3) with q=-1.Comment: 7 A4 page

Killing spinors are Killing vector fields in Riemannian Supergeometry

A supermanifold M is canonically associated to any pseudo Riemannian spin manifold (M_0,g_0). Extending the metric g_0 to a field g of bilinear forms g(p) on T_p M, p\in M_0, the pseudo Riemannian supergeometry of (M,g) is formulated as G-structure on M, where G is a supergroup with even part G_0\cong Spin(k,l); (k,l) the signature of (M_0,g_0). Killing vector fields on (M,g) are, by definition, infinitesimal automorphisms of this G-structure. For every spinor field s there exists a corresponding odd vector field X_s on M. Our main result is that X_s is a Killing vector field on (M,g) if and only if s is a twistor spinor. In particular, any Killing spinor s defines a Killing vector field X_s.Comment: 14 pages, latex, one typo correcte

The Minkowski and conformal superspaces

We define complex Minkowski superspace in 4 dimensions as the big cell inside a complex flag supermanifold. The complex conformal supergroup acts naturally on this super flag, allowing us to interpret it as the conformal compactification of complex Minkowski superspace. We then consider real Minkowski superspace as a suitable real form of the complex version. Our methods are group theoretic, based on the real conformal supergroup and its Lie superalgebra.Comment: AMS LaTeX, 44 page

Mirror symmetry and quantization of abelian varieties

The paper consists of two sections. The first section provides a new definition of mirror symmetry of abelian varieties making sense also over $p$-adic fields. The second section introduces and studies quantized theta-functions with two-sided multipliers, which are functions on non-commutative tori. This is an extension of an earlier work by the author. In the Introduction and in the Appendix the constructions of this paper are put into a wider context.Comment: 24 pp., amstex file, no figure

Quantum Mechanics on the h-deformed Quantum Plane

We find the covariant deformed Heisenberg algebra and the Laplace-Beltrami operator on the extended $h$-deformed quantum plane and solve the Schr\"odinger equations explicitly for some physical systems on the quantum plane. In the commutative limit the behaviour of a quantum particle on the quantum plane becomes that of the quantum particle on the Poincar\'e half-plane, a surface of constant negative Gaussian curvature. We show the bound state energy spectra for particles under specific potentials depend explicitly on the deformation parameter $h$. Moreover, it is shown that bound states can survive on the quantum plane in a limiting case where bound states on the Poincar\'e half-plane disappear.Comment: 16pages, Latex2e, Abstract and section 4 have been revise

Research potential as a basis for innovative development of the region

Purpose of work is to determine an amount of influence from region’s innovative activity on effective usage of current scientific-research potential. Innovative activity of regions in many respects depends on the availability and efficient use of the existing research capacity. The main components of the research capacities in the region are: interest of universities, employers and society in research and development and their implementation in practice; development of research infrastructure; and a focus of higher education on the innovative activity of students; financial and tax support of enterprises engaged in innovative activities, from the stat

Duality for the Jordanian Matrix Quantum Group GLg,h(2)GL_{g,h}(2)

We find the Hopf algebra $U_{g,h}$ dual to the Jordanian matrix quantum group $GL_{g,h}(2)$. As an algebra it depends only on the sum of the two parameters and is split in two subalgebras: $U'_{g,h}$ (with three generators) and $U(Z)$ (with one generator). The subalgebra $U(Z)$ is a central Hopf subalgebra of $U_{g,h}$. The subalgebra $U'_{g,h}$ is not a Hopf subalgebra and its coalgebra structure depends on both parameters. We discuss also two one-parameter special cases: $g =h$ and $g=-h$. The subalgebra $U'_{h,h}$ is a Hopf algebra and coincides with the algebra introduced by Ohn as the dual of $SL_h(2)$. The subalgebra $U'_{-h,h}$ is isomorphic to $U(sl(2))$ as an algebra but has a nontrivial coalgebra structure and again is not a Hopf subalgebra of $U_{-h,h}$.Comment: plain TeX with harvmac, 16 pages, added Appendix implementing the ACC nonlinear ma

Q-Boson Representation of the Quantum Matrix Algebra Mq(3)M_q(3)

{Although q-oscillators have been used extensively for realization of quantum universal enveloping algebras,such realization do not exist for quantum matrix algebras ( deformation of the algebra of functions on the group ). In this paper we first construct an infinite dimensional representation of the quantum matrix algebra $M_q ( 3 )$(the coordinate ring of $GL_q (3))$ and then use this representation to realize $GL_q ( 3 )$ by q-bosons.}Comment: pages 18 ,report # 93-00