Baer cotorsion Pairs

LetR be a unital associative ring and two classes of leftR-modules. In [St3] the notion of a ( ) pair was introduced. In analogy to classical cotorsion pairs, a pair (V,W) of subclasses is called a ( ) pair if it is maximal with respect to the classes and the condition Ext _R ¹ (V, W)=0 for all . In this paper we study pairs whereR = ℤ and is the class of all torsion-free abelian groups andT is the class of all torsion abelian groups. A complete characterization is obtained assumingV=L. For example, it is shown that every pair is singly cognerated underV=L. The author was supported by a DFG grant.     

A question in the theory of saturated formations of finite soluble groups

This paper examines the following question. If and are saturated formations then is defined to be the class of all soluble groups whose belong to . In general is a formation, but need not be a saturated formation. Here the smallest saturated formation containing is studied.     

A class of multipliers for $$\mathcal{W}^ \bot $$

Let denote the class of ergodic probability preserving transformations which are disjoint from every weakly mixing system. Let be the class of multipliers for , i.e. ergodic transformations whose all ergodic joinings with any element of are also in . Fix an ergodic rotationT, a mildly mixing actionS of a locally compact second countable groupG and an ergodic cocycle ϕ forT with values inG. The main result of the paper is a sufficient (and also necessary by [LeP] whenG is countable Abelian andS is Bernoullian) condition for the skew product build fromT, ϕ andS to be an element of . Moreover, the self-joinings of such extensions ofT are described with an application to study semisimple extensions of rotations. Dedicated to Hillel Furstenberg on the occasion of his retirement The first-named author was supported in part by CRDF, grant UM1-2546-KH-03. The second-named author was supported in part by KBN grant 1P03A 03826.     

Integral-valued rational functions on valued fields

Let K be a field and a non-trivial valuation ring of K withm as its maximal ideal. Denote by and the rings of polynomials f∈K[X] and rational functions f∈K(X) resp. such that . We prove that for one variable X we have if and only if the completion of (K, ) is locally compact or algebraically closed. In the second case—i.e. if K is dense in the algebraic closure of (K, )—we even get for any number of variables X=(X₁,...,X_n). This work contains parts of the second author's thesis [Ri] written under the supervision of the first author.     

Blaschke-decomposable convex bodies

For a given centred convex bodyK of ℝ,n≥3, let be the class of all convex bodies with the same projection body asK. The question whetherK can be expressed as a Blaschke average of two non-homothetic bodies from is considered. Necessary and sufficient conditions onK to be Blaschke decomposable in are given. The paper provides also a characterization of the bodiesK such that the Blaschke indecomposable bodies in are dense in itself.     

Factorizations of composition formations

It is proved that, if is a singly generated composition formation, where , then is a composition formation. Translated fromMatematicheskie Zametki, Vol. 65, No. 3, pp. 389–395, March, 1999.     

An example of pointwise non-convergence of iterated conditional expectation operators

Given ∈, we construct a sequence , … of Borel sub-sigma-algebras on the unit interval with the following property. Suppose the identity functionf(x)=x is transformed by successive conditioning on , then , then , Then the lim sup, with respect ton, will exceed (pointwise almost-everywhere) 1−∈ and its lim inf will be less than ∈. The sequence of functions also will fail to converge in the . This contrasts with the long-open conjecture that if all the come from a finite set of sigma-algebras, then the resulting sequence of functions must converge in . J. L. King was partially supported by NSF grant DMS-9112595.     

Invariant convex sets and functions in Lie algebras

We say that an invariant convex coneW in a Lie algebras is elliptic if its interior consists of elliptic elements of . If such a cone exists, then has a compactly embedded Cartan subalgebra. The first main result, of this paper is a characterization of those Lie algebras, which contain elliptic invariant cones. If is an invariant domain in such a cone, then we characterize the invariant locally convex functions onD by their restrictions to where is a compactly embedded Cartan subalgebra.     

On Galois isomorphisms between ideals in extensions of local fields

LetL/K be a totally ramified, finite abelian extension of local fields, let and be the valuation rings, and letG be the Galois group. We consider the powers of the maximal ideal of as modules over the group ring . We show that, ifG has orderp ^m (withp the residue field characteristic), ifG is not cyclic (or ifG has orderp), and if a certain mild hypothesis on the ramification ofL/K holds, then and are isomorphic iffr≡r′ modp ^m . We also give a generalisation of this result to certain extensions not ofp-power degree, and show that, in the casep=2, the hypotheses thatG is abelian and not cyclic can be removed.     

Pseudo algebraically closed fields over rings

We prove that for almost allσ ∈G ℚ the field has the following property: For each absolutely irreducible affine varietyV of dimensionr and each dominating separable rational mapϕ:V→ there exists a point a ∈ such thatϕ(a) ∈ ℤ^r. We then say that is PAC over ℤ. This is a stronger property then being PAC. Indeed we show that beside the fields other fields which are algebraic over ℤ and are known in the literature to be PAC are not PAC over ℤ.     

Exponential generating functions and complexity of Lie varieties

Suppose that % MathType!End!2!1! is a variety of Lie algebras, and letc _n( % MathType!End!2!1!) be the dimension of the linear span of all multilinear words onn distinct letters in the free algebraF( % MathType!End!2!1!,X) of the variety % MathType!End!2!1!. We consider an exponential generating function % MathType!End!2!1!, called the complexity function. The complexity function is an entire function of a complex variable provided the variety of Lie algebras is nontrivial. In this paper we introduce the notion of complexity for Lie varieties in terms of the growth of complexity functions; also we describe what the complexity means for the codimension growth of the variety. Our main goal is to specify the complexity of a product of two Lie varieties in terms of the complexities of multiplicands. The main observation here is thatC( % MathType!End!2!1!),z) behaves like a composition of three functionsC( % MathType!End!2!1!),z), exp(z), andC( % MathType!End!2!1!),z). Partially supported by grant RFFI 96-01-00146; the author is grateful to the University of Bielefeld for hospitality, where he was DAAD-fellow.     

The Hopf algebra Rep U_{q} \widehat{\frak g \frak l}_\infty

We define the Hopf algebra structure on the Grothendieck group of finite-dimensional polynomial representations of in the limitN→∞. The resulting Hopf algebra Rep is a tensor product of its Hopf subalgebras Rep_a ,a ∈ ℂ^×/q^2ℤ. Whenq is generic (resp.,q ² is a primitive root of unity of orderl), we construct an isomorphism between the Hopf algebra Rep _a and the algebra of regular functions on the prounipotent proalgebraic group (resp., ). Whenq is a root of unity, this isomorphism identifies the Hopf subalgebra of Rep _a spanned by the modules obtained by pullback with respect to the Frobenius homomorphism with the algebra generated by the coefficients of the determinant of an element of considered as anl×l matrix over the Taylor series. This gives us an explicit formula for the Frobenius pullbacks of the fundamental representations. In addition, we construct a natural action of the Hall algebra associated to the infinite linear quiver (resp., the cyclic quiver withl vertices) on Rep _a and describe the span of tensor products of evaluation representations taken at fixed points as a module over this Hall algebra.     

Some properties of the absolute galois group of a hilbertian field

LetK be a hilbertian field,G(K) its absolute Galois group. IfK is countable, then for a.a. inG(K) ^e , and there is no intermediate field with . Let ∈G(K) ^e . Then for a.a. in .     

On the representation of differentiations by Hamiltonians,operating from an algebra of local observable spin systems

Let and be algebras of local and quasilocal observable spin systems corresponding to the group Zr, be a differentiation invariant with respect to displacements. The question of representation of D in the form of formal Hamiltonian formed by the displacements of an elementx ε is considered. It is shown that such a representation exists if the condition holds, where means an element obtained from the elements [T_kX,a] by some r-multiple process of summation. Translated from Matematicheskii Zametki, Vol. 21, No. 1, pp. 93–98, January, 1977.     

Zur Topologie der Dixmier-Abbildung

Summary  LetG be the coadjoint group of a finite-dimensional complex Lie algebrag. Forg solvable, the Dixmier-map is known to be a homeomorphism of the orbit space /G onto the space χ of primitive ideals in the enveloping algebra U(G) [6,15]. For , the Dixmier-map is known to be a bijection (and in general not a homeomorphism) with the space χ^l of all completely prime primitive ideals [7, 16]. Here we derive from a result ofW. SOERGEL [18], that this map issheet- wise a homeomorphism onto the image. Here a sheet is a maximal irreducible subset consisting of orbits of a fixed dimension; obviouslyg decomposes into finitely many sheets [3]. The results of this paper hold more generally forg semisimple, if one restricts to a sheet of polarizable orbits, where a Dixmier-map can be defined. Relative to a fixed polarization (a parabolic subalgebra)p ⊂g let I be the annihilator of the generic module induced from p. The „relative enveloping algebra“ ) has been studied e.g. bySOERGEL [19, 18]. Its center Z is described here by a relative Harish-Chandra isomorphism of the normalization with a suitable ring of group invariants (3.2). We study here the extension ofU by . We suggest that this very mild central extension ofU generates good properties and is very suitable for the study of the Dixmier-map (cf.4.3,5.6). In particular, we conjecture in case : Every minimal primitive ideal of is generated by a maximal ideal of the center. This would generalize for a well known theorem ofM. Duflo (casep Borel, where ). AsJ. Dixmier communicated in a letter, the main result here is exactly what he had hoped for when he first introduced a notion of sheets many years ago.

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Added in proof: This conjecture will be proved in a subsequent paper.     

A note on the gl constant ofE \mathop \otimes \limits^ \vee F F

LetX andY be Banach spaces. TFAE (1)X andY do not contain subspaces uniformly isomorphic to (2) The local unconditional structure constant of the space of bounded operatorsL (X*_k,Y _k) tends to infinity for every increasing sequence and of finite-dimensional subspaces ofX andY respectively.     

The 2-length of the hughes subgroup

For a finite groupG and some prime powerp ⁿ , the -subgroup is defined by . Meixner proved that ifG is a finite solvable group and for somen≧1, then the Fitting length of is bounded by 4n. In the following note it is shown that the 2-length of is at mostn. This result cannot be derived from Meixner’s paper, since his result implies only that the 2-length is bounded by 2n.     

Groupwise density and related cardinals

We prove several theorems about the cardinal associated with groupwise density. With respect to a natural ordering of families of nond-ecreasing maps from to, all families of size are below all unbounded families. With respect to a natural ordering of filters on, all filters generated by sets are below all non-feeble filters. If then and . (The definitions of these cardinals are recalled in the introduction.) Finally, some consequences deduced from by Laflamme are shown to be equivalent to .     

Undecidability of the word problem in relatively free rings

An example of a series of varieties of rings with the finite basis property is constructed for which the word problem in the relatively free ring of rankn in the variety is decidable if and only ifn <p. Translated fromMatematicheskie Zametki, Vol. 67, No. 4, pp. 582–594, April, 2000.     

Homogeneous algebraic varieties defined by Jordan pairs

Every Jordan pair defines an algebraic varietyX containing as a dense open subset.X is projective (affine) if and only if is separable (radical). The Picard group ofX is generated by the irreducible factors of the generic norm of . If is separable then the automorphism group ofX is the projective group of .