Lattices of subrepresentations of Lie algebras and their isomorphisms

To obtain the representation (L, R) of Lie algebras over the ring Λ, we construct the lattice of subrepresentations ℒ(L, R). Relations between the algebras L and R and the lattice ℒ(L, R) are studied. It turns out that in some cases the isomorphism of the lattice ℒ(L, R) can be continued so as to obtain a wider sublattice ℒ(LλR) consisting of subalgebras of a semidirect product LλR. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 46, Algebra, 2007.     

Representation fields for quaternionic skew-hermitian forms

Classes of indefinite quadratic forms in a genus are in correspondence with the Galois group of an abelian extension called the spinor class field (Estes and Hsia, Japanese J. Math. 16, 341–350 (1990)). Hsia has proved (Hsia et al., J. Reine Angew. Math. 494, 129–140 (1998)) the existence of a representation field F with the property that a lattice in the genus represents a fixed given lattice if and only if the corresponding element of the Galois group is trivial on F. This far, the corresponding result for skew-hermitian forms was known only in some special cases, e.g., when the ideal (2) is square free over the base field. In this work we prove the existence of representation fields for quaternionic skew-hermitian forms in complete generality.     

The ℒ p spaces

The ℒ _p spaces which were introduced by A. Pełczyński and the first named author are studied. It is proved, e.g., that (i)X is an ℒ _p space if and only ifX* is and ℒ _q space (p ⁻¹+q ⁻¹=1). (ii) A complemented subspace of an ℒ _p space is either an ℒ _p or an ℒ₂ space. (iii) The ℒ _p spaces have sufficiently many Boolean algebras of projections. These results are applied to show thatX is an ℒ_∞ (resp. ℒ₁) space if and only ifX admits extensions (resp. liftings) of compact operators havingX as a domain or range space. We also prove a theorem on the “local reflexivity” of an arbitrary Banach space. This research was partially supported by NSF Grant# 8964.     

On the bicanonical map of a surface section of a threefold

Let (ℳ, ℒ) be a 3-fold of log-general type polarized by a very ample line bundle ℒ. We study the pairs (ℳ, ℒ) in the case when there exists at least one smooth surface Ŝ ∈ |ℒ| such that the bicanonical map associated to |2K_Ŝ| is not birational. As one consequence of our classification we obtain the result:if a smooth projective threefold has non- negative Kodaira dimension, then given any smooth very ample divisor Ŝon the threefold, the bicanonical map associated to |2K_Ŝ|is birational.     

A Note on the Distribution of the Distance from a Lattice

Let ℒ be an n-dimensional lattice, and let x be a point chosen uniformly from a large ball in ℝ ⁿ . In this note we consider the distribution of the distance from x to ℒ, normalized by the largest possible such distance (i.e., the covering radius of ℒ). By definition, the support of this distribution is [0,1]. We show that there exists a universal constant α ₂ that provides a natural “threshold” for this distribution in the following sense. For any ε>0, there exists a δ>0 such that for any lattice, this distribution has mass at least δ on [α ₂−ε,1]; moreover, there exist lattices for which the distribution is tightly concentrated around α ₂ (and so the mass on [α ₂+ε,1] can be arbitrarily small). We also provide several bounds on α ₂ and its extension to other ℓ _p norms. We end with an application from the area of computational complexity. Namely, we show that α ₂ is exactly the approximation factor of a certain natural protocol for the Covering Radius Problem. I. Haviv’s research was supported by the Binational Science Foundation and by the Israel Science Foundation. V. Lyubashevsky’s research was supported by NSF ITR 0313241. O. Regev’s research was supported by an Alon Fellowship, by the Binational Science Foundation, by the Israel Science Foundation, and by the European Commission under the Integrated Project QAP funded by the IST directorate as Contract Number 015848.     

Homotopy equivalences of p-completed classifying spaces of finite groups

We study homotopy equivalences of p-completions of classifying spaces of finite groups. To each finite group G and each prime p, we associate a finite category ℒ _p ^c (G) with the following properties. Two p-completed classifying spaces BG _p ^∧ and BG’ _p ^∧ have the same homotopy type if and only if the associated categories ℒ _p ^c (G) and ℒ _p ^c (G’) are equivalent. Furthermore, the topological monoid Aut(BG _p ^∧) of self equivalences is determined by the self equivalences of the associated category ℒ _p ^c (G). Oblatum 5-VII-2001 & 28-VIII-2002?Published online: 8 November 2002 RID="*" ID="*"C. Broto is partially supported by DGICYT grant PB97–0203. RID="**" ID="**"R. Levi is partially supported by EPSRC grant GR/M7831. RID="***" ID="***"B. Oliver is partially supported by UMR 7539 of the CNRS.     

Some approximation properties of Banach spaces and Banach lattices

The notion of the bounded approximation property = BAP (resp. the uniform approximation property = UAP) of a pair [Banach space, its subspace] is used to prove that if X is a ℒ _∞-space, Y a subspace with the BAP (resp. UAP), then the quotient X/Y has the BAP (resp. UAP). If Q: X → Z is a surjection, X is a ℒ ₁-space and Z is a ℒ _p -space (1 ≤ p ≤ ∞), then ker Q has the UAP. A complemented subspace of a weakly sequentially complete Banach lattice has the separable complementation property = SCP. A criterion for a space with GL-l.u.st. to have the SCP is given. Spaces which are quotients of weakly sequentially complete lattices and are uncomplemented in their second duals are studied. Examples are given of spaces with the SCP which have subspaces that fail the SCP. The results are applied to spaces of measures on a compact Abelian group orthogonal to a fixed Sidon set and to Sobolev spaces of functions of bounded variation on ℝ ⁿ .     

Darstellung durch Spinorgeschlechter ternärer quadratischer formen

Let V be a regular ternary quadratic space over the algebraic number field F, L a lattice on V over the maximal order o of F. A number a, which is represented by all completions L_p, is not necessarily represented by L itself, but only by a lattice in the genus of L. We determine in which cases such a number is not represented by all spinor genera in the genus. Theorem 1 repeats the known [8,3] necessary conditions for this, which show that this behaviour is exceptional. They are sharpened in Theorem 2 to a necessary and sufficient condition in terms of certain groups Θ(L_p, a) (Definition 1). These groups are computed in Theorems 3 and 4 for nondyadic and 2-adip p. Some applications are given in the last section: We give new proofs (and in one case a correction) of results from [5] on the numbers represented by some genera of positive definite ternaries.     

Contractions of infrainvariant systems of subgroups

We create a method which allows an arbitrary group G with an infrainvariant system ℒ(G) of subgroups to be embedded in a group G* with an infrainvariant system ℒ(G*) of subgroups, so that G _α^* ∩G ∈ ℒ(G) for every subgroup G _α^* ∩G ∈ ℒ(G*) and each factor B/A of a jump of subgroups in ℒ(G*) is isomorphic to a factor of a jump in ℒ(G), or to any specified group H. Using this method, we state new results on right-ordered groups. In particular, it is proved that every Conrad right-ordered group is embedded with preservation of order in a Conrad right-ordered group of Hahn type (i.e., a right-ordered group whose factors of jumps of convex subgroups are order isomorphic to the additive group ℝ); every right-ordered Smirnov group is embedded in a right-ordered Smirnov group of Hahn type; a new proof is given for the Holland–McCleary theorem on embedding every linearly ordered group in a linearly ordered group of Hahn type.     

Uniqueness of Nelsons diffusions

Let ℒ≔Δ/2+(∇φ/φ) ·∇ be a generalized Schr?dinger operator or generator of Nelsons diffusion, defined on C ^∞ ₀(D) where φ is a continuous and strictly positive function on an open domain D⊂ℝ ^d such that ∇φ∈L _loc ²(D). Some results are given about the two questions below: (i) Whether does ℒ generate a unique semigroup in L ¹(D, φ² dx)? (ii) Whether the semigroup determined by ℒ is strong Feller? Received: 21 October 1997 / Revised version: 3 September 1998     

Asymptotic resolvent and theorems on generation of semigroups of operators in locally convex spaces

The concept of the asymptotic resolvent of an operator is introduced and used to establish theorems on the generation of semigroups of distributions and operator semigroups of class (ℒ _loc ^p , ℬ) in a locally convex space. Translated from Matematicheskie Zametki, Vol. 22, No. 3, pp. 433–442, September, 1977.     

On zeta functions and families of Siegel modular forms

Let p be a prime, and let G = \textS\textp_g( \mathbbZ ) \Gamma = {\text{S}}{{\text{p}}_g}\left( \mathbb{Z} \right) be the Siegel modular group of genus g. This paper is concerned with p-adic families of zeta functions and L-functions of Siegel modular forms; the latter are described in terms of motivic L-functions attached to Sp _g ; their analytic properties are given. Critical values for the spinor L-functions are discussed in relation to p-adic constructions. Rankin’s lemma of higher genus is established. A general conjecture on a lifting of modular forms from GSp_2m   × GSp_2m to GSp_4m (of genus g = 4 m) is formulated. Constructions of p-adic families of Siegel modular forms are given using Ikeda–Miyawaki constructions.     

Estimates for the closeness of successive convolutions of multidimensional symmetric distributions

Summary Let ξ₁, ξ₂,... be i.i.d random vectors in ℝ ^k with a common distribution ℒ(ξ_i),... = F, i = 1, 2,.... Let S _n = ξ₁+...+ξ _n . We investigate how small is the difference between ℒ(S _n ) and ℒ(S _{n+ m} ) in the case when ξ _i have symmetric distributions.     

Path integrals for quadratic lagrangians on <Emphasis Type="Italic">p</Emphasis>-adic and adelic spaces

Feynman’s path integrals in ordinary, p-adic and adelic quantum mechanics are considered. The corresponding probability amplitudes K(x″, t″; x′, t′) for two-dimensional systems with quadratic Lagrangians are evaluated analytically and obtained expressions are generalized to any finite-dimensional spaces. These general formulas are presented in the form which is invariant under interchange of the number fields ℝ ↔ ℚ _p and ℚ ↔ ℚ _p , p ≠ p′. According to this invariance we have that adelic path integral is a fundamental object in mathematical physics of quantum phenomena.     

Boundedness Properties of Pseudo-Differential and Calderón-Zygmund Operators on Modulation Spaces

In this article, we study the boundedness of pseudo-differential operators with symbols in S _ρ,δ ^m on the modulation spaces M ^p,q . We discuss the order m for the boundedness Op(S _ρ,δ ^m )⊂ℒ(M ^p,q ) to be true. We also prove the existence of a Calderón-Zygmund operator which is not bounded on the modulation space M ^p,q with q≠2. This unboundedness is still true even if we assume a generalized T(1) condition. These results are induced by the unboundedness of pseudo-differential operators on M ^p,q whose symbols are of the class S _1,δ ⁰ with 0<δ<1.      

The concept of the cg-genus of quadratic forms

The concept of cg-genus of quadratic forms is introduced, intermediate between the concepts of genus and spinor genus. This gives the possibility to identify properties of quadratic forms, dependent first of all on the structure of the group of the classes of the ground field. The fundamental properties of cg-genera, analogous in many respects with the properties of spinor genera, are considered. Applications of this concept to various aspects of the problem of the classification of quadratic forms are given. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 160, pp. 170–181, 1987.     

On the Exel Crossed Product of Topological Covering Maps

For dynamical systems defined by a covering map of a compact Hausdorff space and the corresponding transfer operator, the associated crossed product C ^*-algebras C(X)⋊ _α,ℒℕ introduced by Exel and Vershik are considered. An important property for homeomorphism dynamical systems is topological freeness. It can be extended in a natural way to in general non-invertible dynamical systems generated by covering maps. In this article, it is shown that the following four properties are equivalent: the dynamical system generated by a covering map is topologically free; the canonical embedding of C(X) into C(X)⋊ _α,ℒℕ is a maximal abelian C ^*-subalgebra of C(X)⋊ _α,ℒℕ; any nontrivial two sided ideal of C(X)⋊ _α,ℒℕ has non-zero intersection with the embedded copy of C(X); a certain natural representation of C(X)⋊ _α,ℒℕ is faithful. This result is a generalization to non-invertible dynamics of the corresponding results for crossed product C ^*-algebras of homeomorphism dynamical systems.     

Comparison of approximation properties of generalized polynomials and splines

We establish that, for p ∈ [2, ∞), q = 1 or p = ∞, q ∈ [ 1, 2], the classes W _p^rof functions of many variables defined by restrictions on the L _p-norms of mixed derivatives of order r = (r ₁, r ₂, ..., r _m) are better approximated in the L _q-metric by periodic generalized splines than by generalized trigonometric polynomials. In these cases, the best approximations of the Sobolev classes of functions of one variable by trigonometric polynomials and by periodic splines coincide. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 8, pp. 1011–1020, August, 1998.     

On compression of Bruhat-Tits buildings

Consider an affine Bruhat-Tits building Lat _n of type A_n−1 and the complex distance in Lat _n, i.e., the complete system of invariants of a pair of vertices of the building. An element of the Nazarov semigroup is a lattice in the duplicated p-adic space ℚ _p ⁿ ⊕ ℚ _p ⁿ . We investigate the behavior of the complex distance with respect to the natural action of the Nazarov semigroup on the building. Bibliography: 18 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 325, 2005, pp. 163–170.     

Spinor equivalence of quadratic forms

Let f be an integral quadratic form in three or more variables and g any form in the genus of f. There exist an effectively determinable prime p and a form g′, belonging to the proper spinor genus of g, such that g′ is a p-neighbor of f in the graph of f. Using this, an alternative decision procedure for the spinor equivalence of quadratic forms is given.