On Some Extremal Varieties of Associative Algebras

Suppose that F is a field of prime characteristic p and V _p is the variety of associative algebras over F defined by the identities [[x, y], z] = 0 and x ^p = 0 if p > 2 and by the identities [[x, y], z] = 0 and x ⁴ = 0 if p = 2 (here [x, y] = xy ? yx). As is known, the free algebras of countable rank of the varieties V _p contain non-finitely generated T-spaces. We prove that the varieties V _p are minimal with respect to this property.     

On free products in varieties of associative algebras

Varieties of associative algebras over a field of characteristic zero are considered. Belov recently proved that, in any variety of this kind, the Hilbert series of a relatively free algebra of finite rank is rational. At the same time, for three important varieties, namely, those of algebras with zero multiplication, of commutative algebras, and of all associative algebras, a stronger assertion holds: for these varieties, formulas that rationally express the Hilbert series of the free product algebra via the Hilbert series of the factors are well known. In the paper, a system of counterexamples is presented which shows that there is no formula of this kind in any other variety, even in the case of two factors one of which is a free algebra. However, if we restrict ourselves to the class of graded PI-algebras generated by their components of degree one, then there exist infinitely many varieties for each of which a similar formula is valid. Translated fromMatematicheskie Zametki, Vol. 65, No. 5, pp. 693–702, May, 1999.     

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.     

Interpretability Types for Regular Varieties of Algebras

It is proved that for every regular variety V of algebras, an interpretability type [V] in the lattice is primary w.r.t. intersection, and so has at most one covering. Moreover, the sole covering, if any, for [V] is necessarily infinite. For a locally finite regular variety V, [V] has no covering. Cyclic varieties of algebras turn out to be particularly interesting among the regular. Each of these is a variety of n-groupoids (A; f) defined by an identity , where is an n-cycle of degree n 2. Interpretability types of the cyclic varieties form, in , a subsemilattice isomorphic to a semilattice of square-free natural numbers n 2, under taking m n=[m,n] (l.c.m.).     

No Associative PI-Algebra Coincides with Its Commutant

We prove that each (possibly not finitely generated) associative PI-algebra does not coincide with its commutant. We thus solve I. V. L'vov's problem in the Dniester Notebook. The result follows from the fact (also established in this article) that, in every T-prime variety, some weak identity holds and there exists a central polynomial (the existence of a central polynomial was earlier proved by A. R. Kemer). Moreover, we prove stability of T-prime varieties (in the case of characteristic zero, this was done by S. V. Okhitin who used A. R. Kemer's classification of T-prime varieties).     

On the Length of Scales of Computability Potentials for n-Element Algebras

We find the length of scales of computability potentials for n-element algebras. As a consequence, we find the length of the filter generated by the clone of a discriminant function in the lattice of clones of functions on an n-element set.     

Nonuniqueness Sequences for Weighted Algebras of Holomorphic Functions in the Unit Circle

Suppose that is a system of continuous subharmonic functions in the unit disk and is the class of holomorphic functions f in such that log|f(z)| ≤ B _f p _f (z) + C _f , z ∈ , where B _f and C _f are constants and p _f ∈ . We obtain sufficient conditions for a given number sequence Λ = { λ_n} ⊂ to be a subsequence of zeros of some nonzero holomorphic function from , i.e., Λ is a nonuniqueness sequence for .__________Translated from Matematicheskie Zametki, vol. 77, no. 5, 2005, pp. 775–787.Original Russian Text Copyright ©2005 by L. Yu. Cherednikova.     

On the Embedding of Analytic Moufang Loops in Alternative Algebras

Mathematical Notes -     

On Some Sets of Group Functions

Let G be a group, let A be an Abelian group, and let n be an integer such that n –1. In the paper, the sets _n (G,A) of functions from G into A of degree not greater than n are studied. In essence, these sets were introduced by Logachev, Sal'nikov, and Yashchenko. We describe all cases in which any function from G into A is of bounded (or not necessarily bounded) finite degree. Moreover, it is shown that if G is finite, then the study of the set _n (G,A) is reduced to that of the set n(G/O ^p (G),A _p ) for primes p dividing G/G. Here O ^p (G) stands for the p-coradical of the group G, A _p for the p-component of A, and G for the commutator subgroup of G.     

Algebras and Spaces of Dense Constancies

A DC-space (or space of dense constancies) is a Tychonoff space X such that for each f C(X) there is a family of open sets {U _i : i I}, the union of which is dense in X, such that f, restricted to each U _i , is constant. A number of characterizations of DC-spaces are given, which lead to an algebraic generalization of the concept, which, in turn, permits analysis of DC-spaces in the language of archimedean f-algebras. One is led naturally to the notion of an almost DC-space (in which the densely constant functions are dense), and it is shown that all metrizable spaces have this property.     

On Six-Dimensional G2-Submanifolds of Cayley Algebras

It is proved that a generic-type 6-dimensional almost Hermitian submanifold of the algebra of octaves is minimal if and only if it belongs to the Gray--Hervella class G2. This is a maximal strengthening of the well-known result of Gray, who proved the minimality of the 6-dimensional Kähler submanifolds of the Cayley algebra.     

Analyzing the Complexity of Finding Good Neighborhood Functions for Local Search Algorithms

A drawback to using local search algorithms to address NP-hard discrete optimization problems is that many neighborhood functions have an exponential number of local optima that are not global optima (termed L-locals). A neighborhood function η is said to be stable if the number of L-locals is polynomial. A stable neighborhood function ensures that the number of L-locals does not grow too large as the instance size increases and results in improved performance for many local search algorithms. This paper studies the complexity of stable neighborhood functions for NP-hard discrete optimization problems by introducing neighborhood transformations. Neighborhood transformations between discrete optimization problems consist of a transformation of problem instances and a corresponding transformation of solutions that preserves the ordering imposed by the objective function values. In this paper, MAX Weighted Boolean SAT (MWBS), MAX Clause Weighted SAT (MCWS), and Zero-One Integer Programming (ZOIP) are shown to be NPO-complete with respect to neighborhood transformations. Therefore, if MWBS, MCWS, or ZOIP has a stable neighborhood function, then every problem in NPO has a stable neighborhood function. These results demonstrate the difficulty of finding effective neighborhood functions for NP-hard discrete optimization problems.This research is supported in part by the Air Force Office of Scientific Research (F49620-01-1-0007, FA9550-04-1-0110).     

Representations of Some Hopf Algebras Associated to the Symmetric Group S n

We prove that all irreducible representations of the bismash product have Frobenius–Schur indicator +1, where k is an algebraically closed field of characteristic 0. If n = p, a prime, we find all indicators for . We also study more general bismash products. Both authors were supported by NSF grants DMS-07-01291 and DMS-04-01399.     

On the scarcity of contravariant left adjunctions

It is shown that any pair of contravariant mutually left adjoint functors between varieties of algebrasC andD factors through the retractions of these categories onto the complete lattices of epimorphs of their respective initial objects, and is thus fairly degenerate. However, some interesting contravariant left adjoint functors do exist between categories of finitely generated (or otherwise restricted) algebras.This work was done while the author was partly supported by NSF contract DMS 85-02330.Presented by F. E. J. Linton.     

Twisted Bimodules and Hochschild Cohomology for Self-injective Algebras of Class A n , II

Up to derived equivalence, the representation-finite self-injective algebras of class A _n are divided into the wreath-like algebras (containing all Brauer tree algebras) and the Möbius algebras. In Part I (Forum Math. 11 (1999), 177–201), the ring structure of Hochschild cohomology of wreath-like algebras was determined, the key observation being that kernels in a minimal bimodule resolution of the algebras are twisted bimodules. In this paper we prove that also for Möbius algebras certain kernels in a minimal bimodule resolution carry the structure of a twisted bimodule. As an application we obtain detailed information on subrings of the Hochschild cohomology rings of Möbius algebras.     

Kolmogorov Entropy for Classes of Convex Functions

Kolmogorov ε-entropy of a compact set in a metric space measures its metric massivity and thus replaces its dimension which is usually infinite. The notion quantifies the compactness property of sets in metric spaces, and it is widely applied in pure and applied mathematics. The ε-entropy of a compact set is the most economic quantity of information that permits a recovery of elements of this set with accuracy ε. In the present article we study the problem of asymptotic behavior of the ε-entropy for uniformly bounded classes of convex functions in L _p -metric proposed by A.I.   Shnirelman. The asymptotic of the Kolmogorov ε-entropy for the compact metric space of convex and uniformly bounded functions equipped with L _p -metric is ε ^−1/2, ε→0₊.      

On the measure of algebraic independence of values of derivatives of ap-adic modular function

The measure of the relative transcendence for the values of the derivatives of ap-adic Weyl-Lutz modular functionJ(q) at algebraic points is estimated. Translated fromMatematicheskie Zametki, Vol. 61, No. 3, pp. 431–440, March, 1997. Translated by O. V. Sipacheva