Generalized Harish-Chandra Modules: A New Direction in the Structure Theory of Representations

Let be a reductive Lie algebra over C. We say that a -module M is a generalized Harish-Chandra module if, for some subalgebra , M is locally -finite and has finite -multiplicities. We believe that the problem of classifying all irreducible generalized Harish-Chandra modules could be tractable. In this paper, we review the recent success with the case when is a Cartan subalgebra. We also review the recent determination of which reductive in subalgebras are essential to a classification. Finally, we present in detail the emerging picture for the case when is a principal 3-dimensional subalgebra.     

Homological Dimensions of the Algebra Formed by Entire Functions of Elements of a Nilpotent Lie Algebra

This note deals with homological characteristics of algebras of holomorphic functions of noncommuting variables generated by a finite-dimensional nilpotent Lie algebra . It is proved that the embedding of the universal enveloping algebra of into its Arens–Michael hull is an absolute localization in the sense of Taylor provided that      

Quotient Groups of Groups of Certain Classes

For an arbitrary variety of groups and an arbitrary class of groups that is closed on quotient groups, we prove that a quotient group G/N of the group G possesses an invariant system with - and -factors (respectively, is a residually -group) if G possesses an invariant system with - and -factors (respectively, is a residually -group) and N (respectively, N is a maximal invariant -subgroup of the group G).     

The property of having independent basis in semigroup varieties

There exist independently based semigroup varieties and , , such that has no cover in the interval [ ; ].Translated from Algebra i Logika, Vol. 44, No. 1, pp. 81–96, January–February, 2005.     

Derangements and Tensor Powers of Adjoint Modules for \mathfrak{s}\mathfrak{l}_n

We obtain the decomposition of the tensor space as a module for , find an explicit formula for the multiplicities of its irreducible summands, and (when n 2k) describe the centralizer algebra = ( ) and its representations. The multiplicities of the irreducible summands are derangement numbers in several important instances, and the dimension of is given by the number of derangements of a set of 2k elements.     

An Application of U(g)-bimodules to Representation Theory of Affine Lie Algebras

Let be the affine Lie algebra associated to the simple finite-dimensional Lie algebra . We consider the tensor product of the loop -module associated to the irreducible finite-dimensional -module V() and the irreducible highest weight -module L _k,. Then L _k, can be viewed as an irreducible module for the vertex operator algebra M _k,0. Let A(L _k,) be the corresponding -bimodule. We prove that if the -module is zero, then the -module is irreducible. As an example, we apply this result on integrable representations for affine Lie algebras.     

Degree Spectra of Relations on Boolean Algebras

We show that every computable relation on a computable Boolean algebra is either definable by a quantifier-free formula with constants from (in which case it is obviously intrinsically computable) or has infinite degree spectrum.     

Algebras Generated by the Bergman and Anti-Bergman Projections and by Multiplications by Piecewise Continuous Functions

The C*-algebra generated by the Bergman and anti-Bergman projections and by the operators of multiplication by piecewise continuous functions on the Lebesgue space L²(Π) over the upper half-plane is studied. Making use of a local principle, limit operators techniques, and the Plamenevsky results on two-dimensional singular integral operators with coefficients admitting homogeneous discontinuities we reduce the study to simpler C*-algebras associated with points and pairs We construct a symbol calculus for unital C*-algebras generated by n orthogonal projections sum of which equals the unit and by m one-dimensional orthogonal projections. Such algebras are models of local algebras at points z ∈∂Π being the discontinuity points of coefficients. A symbol calculus for the C*- algebra and a Fredholm criterion for the operators are obtained. Finally, a C*-algebra isomorphism between the quotient algebra where is the ideal of compact operators, and its analogue for the unit disk is constructed.     

Trajectory and Global Attractors of Three-Dimensional Navier--Stokes Systems

We construct the trajectory attractor of a three-dimensional Navier--Stokes system with exciting force . The set consists of a class of solutions to this system which are bounded in , defined on the positive semi-infinite interval of the time axis, and can be extended to the entire time axis so that they still remain bounded-in- solutions of the Navier--Stokes system. In this case any family of bounded-in- solutions of this system comes arbitrary close to the trajectory attractor . We prove that the solutions are continuous in t if they are treated in the space of functions ranging in . The restriction of the trajectory attractor to , , is called the global attractor of the Navier--Stokes system. We prove that the global attractor thus defined possesses properties typical of well-known global attractors of evolution equations. We also prove that as the trajectory attractors and the global attractors of the -order Galerkin approximations of the Navier--Stokes system converge to the trajectory and global attractors and , respectively. Similar problems are studied for the cases of an exciting force of the form depending on time and of an external force rapidly oscillating with respect to the spatial variables or with respect to time .     

On a Problem of the Theory of Multiply Local Formations

We describe the -closed n-multiply local formations such that the lattice of all -closed n-multiply local formations between and is Boolean.     

Infinite Independent Systems of Identities of an Associative Algebra over an Infinite Field of Characteristic Two

Let be the variety of associative (special Jordan, respectively) algebras over an infinite field of characteristic 2 defined by the identity ((((x ₁,x ₂),x ₃), ((x ₄,x ₅),x ₆)), (x ₇,x ₈)) = 0 (((x ₁ x ₂ · x ₃)(x ₄ x ₅ · x ₆))(x ₇ x ₈) = 0, respectively). In this paper, we construct infinite independent systems of identities in the variety ( , respectively). This implies that the set of distinct nonfinitely based subvarieties of the variety has the cardinality of the continuum and that there are algebras in with undecidable word problem.     

A family of transitive modular Lie superalgebras with depth one

The embedding theorem is established for Z-graded transitive modular Lie superalgebras g=(?)satisfying the conditions: (i)g0(?)(g-1)and g0-module g-1 is isomorphic to the natural(?)(g-1)-module; (ii)dim g1=2/3n(2n~2 1),where n=1/2dim g-1. In particular,it is proved that the finite-dimensional simple modular Lie superalgebras satisfying the conditions above are isomorphic to the odd Hamiltonian superalgebras.The restricted Lie superalgebras are also considered.     

Uniform Convergence of Trigonometric Series with Rarely Changing Coefficients

We consider the series and whose coefficients satisfy the condition for , where the sequence can be expressed as the union of a finite number of lacunary sequences. The following results are obtained. If as , then the series is uniformly convergent. If for all , then the sequence of partial sums of this series is uniformly bounded. If the series is convergent for and as , then this series is uniformly convergent. If the sequence of partial sums of the series for is bounded and for all , then the sequence of partial sums of this series is uniformly bounded. In these assertions, conditions on the rates of decrease of the coefficients of the series are also necessary if the sequence is lacunary. In the general case, they are not necessary.     

Equational Theories for Classes of Finite Semigroups

It is proved that there exists an infinite sequence of finitely based semigroup varieties such that, for all i, an equational theory for and for the class of all finite semigroups in is undecidable while an equational theory for and for the class of all finite semigroups in is decidable. An infinite sequence of finitely based semigroup varieties is constructed so that, for all i, an equational theory for and for the class of all finite semigroups in is decidable whicle an equational theory for and for the class of all finite semigroups in is not.     

Factorizations of One-Generated Composition Formations

A non-empty formation of finite groups is said to be solubly saturated, or we call it a composition formation, if every finite group G having a normal subgroup N such that belongs to . An intersection of all composition formations containing a given group G is denoted cformG. Conditions are described under which has the form , where .     

A Question in the Theory of Totally Local Formations of Finite Groups

It is proved that all proper totally local subformations of a non one-generated totally local formation of finite groups are one-generated iff coincides with a formation of all soluble -groups, where ||=2.     

Continuous Functors and Duality

Let be an associative ring with identity, and let be the category of left unitary -modules. A complete characterization of continuous additive co- and contravariant functors is given. Such functors are either representable, or equivalent to a tensor product, or trivial ones. The class of categories that are dual to and, therefore, are equivalent to the category of compact right -modules is constructed by purely algebraic means. A canonical category is singled out in this class. A purely algebraic structure that is equivalent to the topology-algebraic structure of compact right -modules is constructed. Algebraic analogs of connection and complete disconnection are given. Bibliography: 6 titles.     

Semisimple Lie Algebras of Differential Operators

Starting from the commutation relations in a complex semisimple Lie algebra , one may obtain a space of vector fields on Euclidean space such that and are isomorphic when is equipped with the usual Lie bracket between vector fields and the isotropy subalgebra of is a Borel subalgebra . Furthermore, one may adjoin to the vector fields in multiplication operators to obtain an -parameter family of distinct presentations of as spaces of differential operators, where is the dual of a Cartan subalgebra. Some of these presentations will preserve a space of polynomials on Euclidean space, and, in fact, all the finite-dimensional representations of can be presented in this way. All of this is carried out explicitly for arbitrary . In doing so, one discovers there is a Lie group of diffeomorphisms of the unipotent subgroup N complementary to B which acts on these presentations and preserves a certain notion of weight.     

Beckman-Quarles type theorems for mappings from $$ \mathbb{R}^n $$ to $$ \mathbb{C}^n $$

Summary. Let We say that preserves the distance d 0 if for each implies Let A _n denote the set of all positive numbers d such that any map that preserves unit distance preserves also distance d. Let D _n denote the set of all positive numbers d with the property: if and then there exists a finite set S _xy with such that any map that preserves unit distance preserves also the distance between x and y. Obviously, We prove: (1) (2) for n 2 D _n is a dense subset of (2) implies that each mapping f from to (n 2) preserving unit distance preserves all distances, if f is continuous with respect to the product topologies on and      

Transversals of Families of Translations of a Planar Convex Compact

Let be a family of translations of a convex compact set such that every two elements of have a common point. Then there exist three points such that each element of contains one of these points. This answers in the affirmative an old question by Grünbaum. Bibliography: 13 titles.