Involutory Quasi-Hopf Algebras

We introduce and investigate the basic properties of an involutory (dual) quasi-Hopf algebra. We also study the representations of an involutory quasi-Hopf algebra and prove that an involutory dual quasi-Hopf algebra with non-zero integral is cosemisimple.     

Three-Selmer groups for elliptic curves with 3-torsion

We consider a specific family of elliptic curves with rational 3-torsion subgroup. We arithmetically define 3-Selmer groups through isogeny and 3-descent maps, then associate the image of the 3-descent maps to solutions of homogeneous cubic polynomials affiliated with the elliptic curve E and an isogenous curve E′. Thanks to the work of Cohen and Pazuki, we have solubility conditions for the homogeneous polynomials. Using these conditions, we give a graphical approach to computing the size of 3-Selmer groups. Finally, we translate the conditions on graphs into a question concerning ranks of matrices and give an upper bound for the rank of the elliptic curve E by calculating the size of the Selmer groups.     

Bounding Ext for modules for algebraic groups, finite groups and quantum groups

Given a finite root system Φ, we show that there is an integer c=c(Φ) such that , for any reductive algebraic group G with root system Φ and any irreducible rational G-modules L, L^′. There also is such a bound in the case of finite groups of Lie type, depending only on the root system and not on the underlying field. For quantum groups, a similar result holds for Extn, for any integer n?0, using a constant depending only on n and the root system. When L is the trivial module, the same result is proved in the algebraic group case, thus giving similar bounded properties, independent of characteristic, for algebraic and generic cohomology. (A similar result holds for any choice of L=L(λ), even allowing λ to vary, provided the p-adic expansion of lambda is limited to a fixed number of terms.) In particular, because of the interpretation of generic cohomology as a limit for underlying families of finite groups, the same boundedness properties hold asymptotically for finite groups of Lie type. The results both use, and have consequences for, Kazhdan–Lusztig polynomials. Appendix A proves a stable version, needed for small prime arguments, of Donkin's tilting module conjecture.     

Galois objects for algebraic quantum groups

The basic elements of Galois theory for algebraic quantum groups were given in the paper ‘Galois Theory for Multiplier Hopf Algebras with Integrals’ by Van Daele and Zhang. In this paper, we supplement their results in the special case of Galois objects: algebras equipped with a Galois coaction by an algebraic quantum group, such that only the scalars are coinvariants. We show how the structure of these objects is as rich as the one of the quantum groups themselves: there are two distinguished weak K.M.S. functionals, related by a modular element, and there is an analogue of the antipode squared. We show how to reflect the quantum group across a Galois object to obtain a (possibly) new algebraic quantum group. We end by considering an example.     

Module twistors for modules of nonlocal vertex algebras

We introduce the notion of module twistor for a module of a nonlocal vertex algebra. The aim of this paper is to use this concept to unify some deformed constructions of modules of nonlocal vertex algebras, such as twisted tensor products and iterated twisted tensor products of modules of nonlocal vertex algebras.     

Differential calculus forq-deformed twistors

A “short” version of the q-deformed differential calculus on the light cone using the twistor representation is proposed. The commutation relations between coordinates and momenta are obtained. The quasi-classical limit gives an exact shape for the off-shell shifting. Translated from Teoreticheskaya i Matematicheskaya Fizika. Vol. 115. No. 2. pp. 177 184. May. 1998.     

Monodromy groups of regular elliptic surfaces

Monodromy in analytic families of smooth complex surfaces yields groups of isotopy classes of orientation preserving diffeomorphisms for each family member X. For all deformation classes of minimal elliptic surfaces with , we determine the monodromy group of a representative X, i.e. the group of isometries of the intersection lattice torsion generated by the monodromy action of all families containing X. To this end we construct families such that any isometry is in the group generated by their monodromies or does not respect the invariance of the canonical class or the spinor norm. Received: 6 June 2000 ; in final form: 26 October 2000 / Published online: 19 October 2001     

Selmer groups of supersingular elliptic curves

The orders of growth of the Selmer groups of supersingular elliptic curves in cyclotomic 2-extensions are established. These orders have the form 2ⁿ/3 or 2ⁿ⁺¹/3 which alternate depending on the parity of n — the number of the level of the cyclotomic extension. This agrees with the hypothetical formulas indicated earlier by A. G. Nasybullin.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Maternaticheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 75, pp. 16–21, 1978.     

Pairings and actions for dynamical quantum groups

Dynamical quantum groups constructed from a FRST-construction using a solution of the quantum dynamical Yang-Baxter equation are equipped with a natural pairing. The interplay of the pairing with *-structures, corepresentations and dynamical representations is studied, and natural left and right actions are introduced. Explicit details for the elliptic U(2) dynamical quantum group are given, and the pairing is calculated explicitly in terms of elliptic hypergeometric functions. Dynamical analogues of spherical and singular vectors for corepresentations are introduced.     

Elementary number theory in superstrings,loop quantum mechanics,twistors and E-infinity high energy physics

The role of elementary number theory in high energy physics is discussed in some details.     

Gauss decomposition for quantum groups and supergroups

The Gauss decompositions of quantum groups related to classical Lie groups and supergroups are considered by the elementary algebraic and R-matrix methods. The commutation relations between the generators of the new basis introduced by the decomposition are described in detail. It is shown that it is possible to reduce a number of independent generators to the dimension of the related classical group. The symplectic quantum group Sp_q(2) and supergroups GL_q(1, 1) and GL_q(2, 1) are considered as examples. Bibliography: 73 titles. In memory of Victor Nikolaevich Popov Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 224, 1995, pp. 155–177. Translated by E. V. Damanskinskii.