Invariant polynomials under the maximal unipotent subgroup of a classical group

Let N be a maximal unipotent subgroup of a classical complex Lie group G, whose Lie algebra we denote by g. Inside the ring of N-invariant polynomials S′(g)^N we consider the subring generated by the polynomials of weight e^{m(Λ + Λ*)}, where Λ and Λ^* are the highest weights corresponding to the natural representation of g and to its dual respectively, and m ε Z.We prove that this subring is a polynomial ring and we explicitly give a set of generators.     

Criteria for the primitivity of a system of elements of a free metabelian group

We state the criteria for the complementability of an arbitrary system of elements to a set of free generators of a free metabelian group.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, Nos. 7 and 8, pp. 996–1002, July–August, 1991.     

Filtration of the group of principal units of a local function field as a Galois module

Let k be the field of formal power series in one variable over a finite field of constants of characteristic p and K/k be a completely ramified extension of degree p. For the group E₁ of principal units of the field K, considered as a Galois module, generators are constructed effectively. The operator structure (as a Galois module) of the subgroups E_m generating a natural filtration of the group E₁ is determined.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 103, pp. 96–99, 1980.     

A group of exponent 8

A group of exponent 8 with two generators one of which has order 4 and the other has order 2 is studied. The question of finiteness of such a group is reduced to the question of finiteness over a field of two elements of a Lie algebra with two generators and three defining relations of a very simple form.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 74, pp. 164–165, 1978.     

Cryptanalysis of a Matrix-based MOR System

We cryptanalyze a recently proposed matrix-based MOR cryptosystem. The security of the system depends on the difficulty of solving the following discrete logarithm problem: given an inner automorphism φ of SL(d, 𝔽 _q ) and φ ^a (each given in terms of their images on generators of SL(d, 𝔽 _q )), find a. We show that this problem can be reduced to a small number of similar problems in quotients of polynomial rings and solved in subexponential-time.     

The classical analog of intertwining operators in the relativistic Hamiltonian mechanics of a system of particles

In the context of nonquantum Hamiltonian formalism of the relativistic theory of direct interaction we construct a canonical transformation of the collective variables of center of mass type which transforms the canonical generators of the Poincaré algebra in one form of dynamics into the corresponding generators in another form of dynamics.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 32, 1990, pp. 62–65.     

Canonical decomposition in the group of points of a formal Lubin-Tate group

In the group of points of a formal Lubin-Tate group over a local field there is constructed a canonical decomposition needed for giving an explicit formula for the Hilbert symbol. Here all arguments are independent of the parity of the characteristic of the residue field.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 103, pp. 52–57, 1980.     

Structure of the group of principal units in a cyclic ρ-extension of a local field

The structure of the group of principal units and the multiplicative group of cyclic p-extensions of an irregular local field are studied in terms of generators and defining relations.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 57, pp. 51–63, 1976.     

Asymptotic analysis of a volterra integral equation

A uniformly valid aymptotic solution is obtained for a class of perturbed Volterra integral equations, in which a naive expansion breaks down as t → ∞. The procedure used is an adaption of the formal methodology presented in [1] for the construction of a uniform asymptotic solution to Volterra equations which possess a boundary layer near t = 0.     

Hopf coactions on commutative algebras generated by a quadratically independent comodule

Let 𝒜 be a commutative unital algebra over an algebraically closed field k of characteristic ≠2, whose generators form a finite-dimensional subspace V, with no nontrivial homogeneous quadratic relations. Let 𝒬 be a Hopf algebra that coacts on 𝒜 inner-faithfully, while leaving V invariant. We prove that 𝒬 must be commutative when either: (i) the coaction preserves a non-degenerate bilinear form on V; or (ii) 𝒬 is co-semisimple, finite-dimensional, and char(k) = 0.     

On the nilpotency class of a multiplicative group of a modular group algebra of a dihedral 2-group

It is proved that the wreath product of a second-order group and the commutant of a dihedral group is imbedded into a multiplicative group of a modular group algebra of a dihedral group of order 2 ⁿ . This implies that the nilpotency class of the multiplicative group is equal to 2 ^n–2, i.e., to the order of the commutant of the dihedral group.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 1, pp. 39–45, January, 1995.     

Abelian torsion groups with a countably compact group topology

Comfort and Remus [W.W. Comfort, D. Remus, Abelian torsion groups with a pseudocompact group topology, Forum Math. 6 (3) (1994) 323–337] characterized algebraically the Abelian torsion groups that admit a pseudocompact group topology using the Ulm–Kaplansky invariants.We show, under a condition weaker than the Generalized Continuum Hypothesis, that an Abelian torsion group (of any cardinality) admits a pseudocompact group topology if and only if it admits a countably compact group topology. Dikranjan and Tkachenko [D. Dikranjan, M. Tkachenko, Algebraic structure of small countably compact Abelian groups, Forum Math. 15 (6) (2003) 811–837], and Dikranjan and Shakhmatov [D. Dikranjan, D. Shakhmatov, Forcing hereditarily separable compact-like group topologies on Abelian groups, Topology Appl. 151 (1–3) (2005) 2–54] showed this equivalence for groups of cardinality not greater than .We also show, from the existence of a selective ultrafilter, that there are countably compact groups without non-trivial convergent sequences of cardinality κ^ω, for any infinite cardinal κ. In particular, it is consistent that for every cardinal κ there are countably compact groups without non-trivial convergent sequences whose weight λ has countable cofinality and λ>κ.     

Metabelian group of exponent 9 with two generators

We prove the theorem: the order of a metabelian group of exponent 9 with two generators does not exceed 3⁷¹. Its index of nilpotency is not greater than 16.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 103, pp. 124–131, 1980.     

On an action of the group ring of a group of field automorphisms

Novosibirsk. Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 33, No. 6, pp. 162–168, November–December, 1992.     

On the divisor class group of double solids

For a double solid V→ℙ_3> branched over a surface B⊂ℙ₃(ℂ) with only ordinary nodes as singularities, we give a set of generators of the divisor class group in terms of contact surfaces of B with only superisolated singularities in the nodes of B. As an application we give a condition when H ^* (˜V , ℤ) has no 2-torsion. All possible cases are listed if B is a quartic. Furthermore we give a new lower bound for the dimension of the code of B. Received: 16 November 1998     

Diameter of a direct power of a finite group

We present two conjectures concerning the diameter of a direct power of a finite group. The first conjecture states that the diameter of Gⁿ with respect to each generating set is at most n(|G|?rank(G)); and the second one states that there exists a generating set 𝒜, of minimum size, for Gⁿ such that the diameter of Gⁿ with respect to 𝒜 is at most n(|G|?rank(G)). We will establish evidence for each of the above mentioned conjectures.     

Rigorous Formulation of a 2D Conformal Model in the Fock–Krein Space: Construction of the Global Op J *-Algebra of Fields and Currents

We use the formalism of the 2D massless scalar field model in an indefinite space of the Fock–Krein type as a basis for constructing a rigorous formulation of 2D quantum conformal theories. We show that the sought construction is a several-stage procedure whose central block is the construction of a new type of representation of the Virasoro algebra. We develop the first stage of this procedure, which is to construct a special global algebra of fields and currents generated by exponential generators. We obtain a system of commutation relations for the Wick-squared currents used in the definition of the Virasoro generators. We prove the existence of Wick exponentials of the current given by operator-valued generalized functions; the sought global algebra is rigorously defined as the algebra of current and field, Wick and normal exponentials on a common dense invariant domain in a Fock–Krein space.