Table of contents

The finite-dimensional semisimple Hopf algebras over an algebraically closed field are classified under the assumption that the algebra has a unique irreducible representation of dimension greater than 1.

Bibliography: 9 titles.

The 2-cohomology group is determined for the finite simple orthogonal group , where is odd, with coefficients in the natural module. For this group is trivial, and for it is isomorphic to . Thus Küsefoglu's result is corrected.

Bibliography: 5 titles.

The structure of Lie groups acting transitively on the direct product of a circle and an even-dimensional sphere is described. For products of two spheres of dimension >1 a similar problem has already been solved by other authors. The minimal transitive Lie groups on and are also indicated.

As an application of these results, the structure of the automorphism group of one class of geometric structures, generalized quadrangles (a special case of Tits buildings) is considered. A conjecture put forward by Kramer is proved: the automorphism group of a connected generalized quadrangle of type (1,2m) always contains a transitive subgroup that is the direct product of a compact simple Lie group and a one-dimensional Lie group.

Bibliography: 16 titles.

We consider algebraic functions satisfying equations of the following form:

(1)

Here , , and is a function of the complex variables . Solutions of such algebraic equations are known to satisfy holonomic systems of linear differential equations with polynomial coefficients. In this paper we investigate one such system, which was introduced by Mellin. The holonomic rank of this system of equations and the dimension of the linear space of its algebraic solutions are computed. An explicit base in the solution space of the Mellin system is constructed in terms of roots of (1) and their logarithms. The monodromy of the Mellin system is shown to be always reducible and several results on the factorization of the Mellin operator in the one-variable case are presented.

Bibliography: 18 titles.

An asymptotic formula as for the solution of the ordinary differential Abel's equation of the first kind , which is uniform in the x-variable, is constructed and substantiated.

Bibliography: 13 titles.

Givental's theorem for complete intersections in smooth toric varieties is generalized to Fano varieties. The Gromov-Witten invariants are found for Fano varieties of dimension ≥3 that are complete intersections in weighted projective spaces or singular toric varieties. A generalized Riemann-Roch equation is also obtained for such varieties. As a consequence, the counting matrices of smooth Fano threefolds with Picard group  and anticanonical degrees 2, 8, and 16 are calculated.

Bibliography: 29 titles.

The existence of a -solution of the homogeneous generalized Wiener-Hopf equation

is proved, where is a probability distribution of recurrent type in . Asymptotic properties of this solution are established.

Bibliography: 10 titles.

Let be a commutative unital Banach algebra with infinite spectrum. Then by Helemskiĭ's global dimension theorem the global homological dimension of is strictly greater than one. This estimate has no analogue for abstract algebras or non-normable topological algebras. It is proved in the present paper that for every unital Banach algebra the global homological dimensions and the homological bidimensions of the Banach algebras and  (assuming certain restrictions on ) are related by and . Thus, a partial extension of Helemskiĭ's theorem to tensor products is obtained.

Bibliography: 28 titles.