Universal (Co) Homology Theories

It is shown that the homology and cohomology theories on separable C*–algebras given by nonstable E–theory are the universal such theories. By specializing to Abelian C*–algebras, we obtain a family of extraordinary Steenrod homology and cohomology theories on pointed compact metric spaces which are the universal such theories in the same way. For each of the extraordinary Steenrod (co)homology theories considered, we describe an –spectrum which represents the theory.     

Finitistic dimension and Igusa–Todorov algebras

The notion of Igusa–Todorov algebras is introduced in connection with the (little) finitistic dimension conjecture, and the conjecture is proved for those algebras. Such algebras contain many known classes of algebras over which the finitistic dimension conjecture holds, e.g., algebras with the representation dimension at most 3, algebras with radical cube zero, monomial algebras and left serial algebras, etc. It is an open question whether all artin algebras are Igusa–Todorov. We provide some methods to construct many new classes of (2-)Igusa–Todorov algebras and thus obtain many algebras such that the finitistic dimension conjecture holds. In particular, we show that the class of 2-Igusa–Todorov algebras is closed under taking endomorphism algebras of projective modules. Hence, if all quasi-hereditary algebras are 2-Igusa–Todorov, then all artin algebras are 2-Igusa–Todorov by [V. Dlab, C.M. Ringel, Every semiprimary ring is the endomorphism ring of a projective module over a quasihereditary ring, Proc. Amer. Math. Soc. 107 (1) (1989) 1–5] and have finite finitistic dimension.     

nvolutorial Płonka sums

The construction of the sum of a direct (semilattice ordered) system of algebras introduced by J. Plonka – later known as the Plonka sum – is one of the most important methods of composition in universal algebra, having a number of applications in different algebraic theories, such as semigroup theory, semiring theory, etc. In this paper we present a more general way for constructing algebras with involution, that is, algebraic systems equipped with a unary involutorial operation which is at the same time an antiautomorphism of the underlying algebra. It is the sum – involutorial Plonka sum, as we call it – of an involution semilattice ordered system of algebras. We investigate its basic properties, as well as the problem of its subdirect decomposition.     

Free expansions of the intersection of manifolds of universal algebras

The theory of free expansions in the intersection of a finite number of manifolds of universal algebras satisfying a given set of conditions (for example, in the manifold of all groups) is extended to the case of the intersection of any set of manifolds of universal algebras.Translated from Matematicheskie Zametki, Vol. 7, No. 5, pp. 551–562, May, 1970.     

Foliations

The survey is based on works on the theory of foliations reviewed in RZhMatematika during 1970–1979. The basic topics are the classification of foliations, characteristic classes, the qualitative theory of foliations (holonomy, growth of leaves, etc.), and special classes of foliations (compact foliations, Riemannian foliations, etc.).Translated from Itogi Nauki i Tekhniki, Seriya Algebra, Topologiya, Geometriya, Vol. 18, pp. 151–213, 1981.     

Cohomology of infinite-dimensional Lie algebras and characteristic classes of foliations

A theory of characteristic classes of foliations is discussed which is based on the cohomology of infinite-dimensional Lie algebras. One of the original results of the article is a theorem stating that the characteristic classes are nontrivial.Translated from Sovremennye Problemy Matematiki, Itogi Nauki i Tekhniki, Vol. 10, pp. 179–285, 1978.     

Semilattices of Definable Subalgebras

In issues bearing on the structure of universal algebras , derived structures, such as automorphism groups Aut , subalgebra lattices Sub , congruence lattices Con , etc., play an important part. On the other hand, in studying universal algebras by the means of model theory, of crucial importance is the question asking which elements of the derived structures under examination are expressible by one or other formulas in the elementary language. Problems concerning the interrelationship of algebras and their derived structures are treated for subalgebras of universal algebras.Supported by RFBR grant No. 02-01-00258.__________Translated from Algebra i Logika, Vol. 44, No. 4, pp. 474–482, July–August, 2005.     

Semisimple classes and lower radicals of topological nonassociative algebras

In this article semisimple classes of topological algebras are characterized in a series of varieties of rings (in particular, all subvarieties of the varieties of alternative and Jordan algebras). Characterizations of semisimple classes of hereditary radicals are obtained, and the heredity of semisimple classes of radicals is demonstrated. It is proved that the construction of lower radicals stabilizes on the first infinite ordinal in the varieties of topological algebras considered.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 14, pp. 250–261, 1989.     

Congruence-distributive varieties of algebras

A survey of results on congruence-distributive varieties of algebras, concentrated on two directions: 1) the construction of algebras of congruence-distributive varieties and the structure of the variety itself, 2) questions connected with formalized (equational, elementary, etc.) theories of such varieties. In addition the survey includes papers connected with congruence-distributive varieties of algebras reviewed in RZh Matematika during the period 1976–1985.Translated from Itogi Nauki i Tekhniki, Seriya Algebra, Topologiya, Geometriya, Vol. 26, pp. 45–83, 1988.     

Lie groups

The survey deals with the investigations reviewed inReferativnyi Zhurnal Matematika between 1977–1981. In the survey there are reflected the investigations on the structure of Lie groups and Lie algebras, on their finite-dimensional linear representations and universal enveloping algebras, on the theory of invariants and Lie groups of transformations, and also on continuous and discrete subgroups of Lie groups.Translated from Itogi Nauki i Tekhniki, Seriya Algebra, Topologiya, Geometriya, Vol. 20, No. 153–192, 1982.     

Nonassociative rings

Results in the theory of nonassociative rings and related directions reviewed in RZhMatematika during the period 1972–1978 are discussed. Special attention is given to infinite-dimensional Lie algebras, Jordan algebras, alternative rings, Mal'tsev algebras, and varieties, representations, and radicals of nonassociative rings.Translated from Itogi Nauki i Tekhniki, Seriya Algebra, Topologiya, Geometriya, Vol. 18, pp 3–72 1981.     

Formations of algebras with complemented subformations

Formations of universal algebras with systems of complemented subformations are considered. In particular, a presentation of the Mal'tsev formations, all subformations of which are complemented, is obtained.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, Nos. 7 and 8, pp. 1008–1012, July–August, 1991.     

Duality of algebras with an involution and generalized shift operators

A survey is given of works in which L. S. Pontryagin's duality principle is extended to various classes of, generally speaking, noncommutative, generalized shift operators. The discussion is carried out with the use of the theory of Kac algebras and its generalizations. A number of problems are formulated.Translated from Itogi Nauki i Tekhniki, Seriya Matematicheskii Analiz, Vol. 24, pp. 165–205, 1986.     

Radical classes of algebras with B-action

We initiate the radical theory of algebras with B-action where B is a fixed Boolean ring. We consider lattices of classes of algebras defined in terms of ideals of B. In two special cases (universal classes of -groups with B-action and idempotent algebras with B-action), these ideal-defined classes are sublattices of the lattice of radicals, and we characterise semisimplicity in such cases. Received February 2, 1998; accepted in final form June 11, 1998.     

Gröbner–Shirshov Bases for Universal Enveloping Conformal Algebras of Simple Conformal Lie Superalgebras of Type WN

For simple conformal Lie superalgebras of type W_N, Gröbner–Shirshov bases of their universal enveloping associative conformal algebras are found. The universal enveloping algebras considered correspond to a minimal locality function for which there is an injective embedding.     

Duality for crossed products of von Neumann algebras

We present the theory of duality for Kats algebras and crossed products of von Neumann algebras and Kats algebras. We give a series of examples.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki (Noveishie Dostizheniya), Vol. 27, pp. 33–65, 1985.     

The weak braided Hopf algebra structure of some Cayley–Dickson algebras

It has been shown by Albuquerque and Majid that a class of unital k-algebras, not necessarily associative, obtained through the Cayley–Dickson process can be viewed as commutative associative algebras in some suitable symmetric monoidal categories. In this note we will prove that they are, moreover, commutative and cocommutative weak braided Hopf algebras within these categories. To this end we first define a Cayley–Dickson process for coalgebras. We then see that the k-vector space of complex numbers, of quaternions, of octonions, of sedenions, etc. fit to our theory, hence they are all monoidal coalgebras as well, and therefore weak braided Hopf algebras.     

On self-adjoint representations of the sklyanin algebras

Self-adjoint representations in Hilbert space of a family of quadratic algebras constructed by E. K. Sklyanin are studied. Complete sets of classes of unitary equivalence of irreducible representations are obtained for algebras that correspond to second-order points of elliptic curves.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 11, pp. 1567–1574, November, 1991.     

Equational classes of Boolean functions via the HSP Theorem

A variant of a theorem of Ekin, Foldes, Hammer and Hellerstein concerning equational characterizations of Boolean function classes is proved using methods of universal algebra. The proof is not constructive but establishes a direct connection with the Birkhoff-Tarski HSP Theorem and the theory of equational classes of universal algebras. Received May 12, 1999; accepted in final form August 14, 2000.     

A characterization of congruence permutable locally finite varieties

A variety of universal algebras is said to be congruence permutable if for every algebra A of and every pair of congruences α, β from A we have αβ = βα. We show that if is locally finite (i.e., every finitely generated member of is finite) then congruence permutability is equivalent to a local property of the finite members of , expressible in the language of tame congruence theory. This answers a question of R. McKenzie and D. Hobby.