Types of action of Lie ε-algebras

We introduce the concept of a type of action. Various types of group actions and actions of Lie ε-algebras are examined. The main result is the classification of types of action F of Lie ε-algebras with the property that for all Lie ε-algebras L, F(L), as an algebra, is the bozonization of L. Supported by RFFR grant No. 93-01-16171 and by ISF grants RPS000 and RPS300. Translated fromAlgebra i Logika, Vol. 35, No. 4, pp. 468–475, July–August, 1996.     

Primitive algebraic relations of skew derivations of prime rings

In [1], the question was posed as to whether or not all algebraic relations of skew derivations of prime rings follow from primitive algebraic relations. Here we argue to obtain a negative answer to a milder question, and namely, an example is constructed in which a pointed Hopf algebra H (generated as an algebra with unity by its relatively primitive elements) acts trivially on the generalized centroid C of a prime ring R, but not all algebraic relations of skew derivations (corresponding to relatively primitive elements in H) follow from primitive algebraic ones. The R in the counterexample is a free associative C-algebra. Supported by ISF grant No. RPS300 and by RFFR grant No. 95-01-01356a. Translated from Algebra i Logika, Vol. 36, No. 4, pp. 407–421, July–August, 1997.     

The rings of quotients of free algebras

It is proved that a symmetric Utumi ring of quotients, U, of a free associative (noncommutative) algebra F(X) with unity coincides with the algebra itself, U=F(X). From this, we obtain a similar statement concerning a symmetric Martindale ring of quotients, Q(F(X))=F(X), which is well known. In addition, it is shown that a left Martindale ring of quotients, F(X)_F, of a free algebra is a prime algebra and, moreover, every homogeneous element in a free algebra has the right inverse in F(X)_F but does not have the left one (unless, of course, r belongs to an underlying field). Since a left Utumi ring of quotients and a left Martindale ring of quotients for a free algebra both appear prime, an interesting question arises as to whether or not they coincide. Supported by RFFR grant No. 95-01-01356 and by ISF grant RPS000-RPS300. Translated fromAlgebra i Logika, Vol. 35, No. 6, pp. 655–662, November–December, 1996.     

Two-dimensional Meixner Random Vectors of Class ${\mathcal{M}}_{L}$

The paper is divided into two parts. In the first part we lay down the foundation for defining the joint annihilation–preservation–creation decomposition of a finite family of not necessarily commutative random variables, and show that this decomposition is essentially unique. In the second part we show that any two, not necessarily commutative, random variables X and Y for which the vector space spanned by the identity and their annihilation, preservation, and creation operators equipped with the bracket given by the commutator forms a Lie algebra are equivalent up to an invertible linear transformation to two independent Meixner random variables with mixed preservation operators. In particular, if X and Y commute, then they are equivalent up to an invertible linear transformation to two independent classic Meixner random variables. To show this we start with a small technical condition called “non-degeneracy”.     

Hochschild cohomologies for associative conformal algebras

We introduce the concept of Hochschild cohomologies for associative conformal algebras. It is shown that the second cohomology group of a conformal Weyl algebra with values in any bimodule is trivial. As a consequence, we derive that the conformal Weyl algebra is segregated in any extension with nilpotent kernel. Supported by RFBR grant No. 05-01-00230 and via SB RAS Integration project No. 1.9. __________ Translated from Algebra i Logika, Vol. 46, No. 6, pp. 688–706, November–December, 2007.     

A local–global principle for algebras with involution and hermitian forms

 Weakly hyperbolic involutions are introduced and a proof is given of the following local–global principle: a central simple algebra with involution of any kind is weakly hyperbolic if and only if its signature is zero for all orderings of the ground field. Also, the order of a weakly hyperbolic algebra with involution is a power of two, this being a direct consequence of a result of Scharlau. As a corollary an analogue of Pfister's local–global principle is obtained for the Witt group of hermitian forms over an algebra with involution. Received: 29 October 2001; in final form: 9 August 2002 / Published online: 16 May 2003 Mathematics Subject Classification (2000): 16K20, 11E39     

Derivation of prime rings of positive characteristic

Let L be a finite-dimensional differential Lie algebra acting on a prime ring R and let the inner part {ie49-1} of L be quasi-Frobenius. Then a constant ring R^L is prime iff {ie49-2} is a differentially simple ring. A ring of constants is semiprime iff {ie49-3} is a direct sum of differentially simple rings, and the prime dimension of a constant ring is equal to the number of differentially simple summands {ie49-4}. The Galois closure of L is obtained from L by adding all the inner derivations of a symmetric Martindale quotient ring which agree with elements from {ie49-5}. Supported by RFFR grant No. 93-01-16171 and by ISF grant RPS000-RPS300. Translated fromAlgebra i Logika, Vol. 35, No. 1, pp. 88–104, January–February, 1996.     

Groups of formal power series are fully orderable

We construct an example of a fully orderable group that is not locally solvable. It is also shown that a free group is embedded in a fully orderable group. To meet these ends, use is made of a group of invertible formal power series with zero free term under composition. Supported by RFFR grant No. 96-01-00088. Translated fromAlgebra i Logika, Vol. 37, No. 3, pp. 301–319, May–June, 1998.     

Nil-elements of index 2 in Mal’tsev algebras

It is proved that a Mal'tsev algebra over an associative commutative ring with 1, which contains 1/6 and is generated by a finite tuple of nil-elements of index 2, is nilpotent, and that an ideal of the Mal'tsev algebra over a field of characteristic 0, generated by nil-elements of index 2, is locally solvable. Supported by RFFR grant No. 96-01-01511. Translated fromAlgebra i Logika, Vol. 37, No. 3, pp. 358–373, May–June, 1998.     

Mal’tsevh 0-algebras

Let Φ be an associative commutative ring with unity, 1/6 ∈ Φ, write A for a Mal’tsev algebra over Φ, suppose that on A, the function h(y, z, t, x, x)=2[{yz, t, x}x+{yx, z, x}t], where {x, y, z}=(xy)z−(xz)y+2x(yz), is defined, and assume that H(A) is a fully invariant ideal of A generated by the function h. The algebra A satisfying an identity h(y, z, x, x, x)=0 [h(y, z, t, x, x)=0] is called a Mal’tsev h₀-algebra (h-algebra). We prove that in any Mal’tsev h₀-algebra, the inclusion H(A)·A₂⊆Ann A holds withAnnA the annihilator of A. This means that any semiprime h₀-algebra A is an h-algebra. Every prime h₀-algebra A is a central simple algebra over the quotient field Λ of the center of its algebra of right multiplications, R(A), and is either a 7-dimensional non-Lie algebra or a 3-dimensional Lie algebra over Λ. Supported by RFFR grant No. 94-01-00381-a. Translated fromAlgebra i Logika, Vol. 35, No. 2, pp. 214–227, March–April, 1996.     

Complexity of Boolean algebras and their Scott rank

We deal with problems associated with Scott ranks of Boolean algebras. The Scott rank can be treated as some measure of complexity of an algebraic system. Our aim is to propound and justify the procedure which, given any countable Boolean algebra, will allow us to construct a Boolean algebra of a small Scott rank that has the same natural algebraic complexity as has the initial algebra. In particular, we show that the Scott rank does not always serve as a good measure of complexity for the class of Boolean algebras. We also study into the question as to whether or not a Boolean algebra of a big Scott rank can be decomposed into direct summands with intermediate ranks. Examples are furnished in which Boolean algebras have an arbitrarily big Scott rank such that direct summands in them either have a same rank or a fixed small one, and summands of intermediate ranks are altogether missing. This series of examples indicates, in particular, that there may be no nontrivial mutual evaluations for the Scott and Frechet ranks on a class of countable Boolean algebras. Supported by RFFR grant No. 99-01-00485, by a grant for Young Scientists from SO RAN, 1997, and by the Federal Research Program (FRP) “Integration”. Translated fromAlgebra i Logika, Vol. 38, No. 6, pp. 643–666, November–December, 1999.     

Jordan bialgebras and their relation to Lie bialgebras

We develop the notion of Jordan bialgebras and study the way in which such are related to Lie bialgebras. In particular, it is shown that if a Lie algebra L(J) obtained from a Jordan algebra J by applying the Kantor-Koecher-Tits construction admits the structure of a Lie bialgebra, under some natural constraints, then, J permits the structure of a Jordan algebra. Supported by RFFR grant No. 95-01-01356 and by ISF grant No. RB 6300. Translated fromAlgebra i Logika, Vol. 36, No. 1, pp. 3–25, January–February, 1997.     

Novikov-Poisson algebras and associative commutative derivation algebras

We describe Novikov-Poisson algebras in which a Novikov algebra is not simple while its corresponding associative commutative derivation algebra is differentially simple. In particular, it is proved that a Novikov algebra is simple over a field of characteristic not 2 iff its associative commutative derivation algebra is differentially simple. The relationship is established between Novikov-Poisson algebras and Jordan superalgebras. Supported by RFBR (grant No. 05-01-00230), by SB RAS (Integration project No. 1.9), and by the Council for Grants (under RF President) and State Aid of Leading Scientific Schools (project NSh-344.2008.1). __________ Translated from Algebra i Logika, Vol. 47, No. 2, pp. 186–202, March–April, 2008.     

Simple iterative algebras

Based on the notion of an ideal of an iterative algebra, developed earlier, we introduce the concept of a simple iterative algebra of functions of a k-valued logic, that is, an algebra freed of proper ideals. It is found out that every simple algebra is contained in a maximal algebra and is naturally associated with some permutation group on a finite set. We specify a number of conditions on a permutation group under which a corresponding algebra is said to be maximal and simple. This allows us to completely list the maximal simple algebras for the case where k<-4. Translated fromAlgebra i Logika, Vol. 37, No. 4, pp. 460–477, July–August, 1998.     

Exponent of a congruence-subgroup of a finite commutative chain ring

The exponent of a congruence-subgroup of the group of invertible matrices and the maximal order of an invertible matrix over the finite commutative wildly ramified chain ring are evaluated. This problem was previously considered in [1] where upper bounds for the exponent of a congruence-subgroup were derived. Translated fromAlgebra i Logika, Vol. 36, No. 6, pp. 657–674, November–December, 1997.     

Non-denseness of factorable matrix functions

It is proved that for certain algebras of continuous functions on compact abelian groups, the set of factorable matrix functions with entries in the algebra is not dense in the group of invertible matrix functions with entries in the algebra, assuming that the dual abelian group contains a subgroup isomorphic to Z³. These algebras include the algebra of all continuous functions and the Wiener algebra. More precisely, it is shown that infinitely many connected components of the group of invertible matrix functions do not contain any factorable matrix functions, again under the same assumption. Moreover, these components actually are disjoint with the subgroup generated by the triangularizable matrix functions.     

The Kantor-Koecher-Tits construction for Jordan coalgebras

The relationship between Jordan and Lie coalgebras is established. We prove that from any Jordan coalgebra 〈L(A), Δ〉, it is possible to construct a Lie coalgebra 〈L(A), Δ_L〉. Moreover, any dual algebra of the coalgebra 〈L(A), Δ_L〉 corresponds to a Lie algebra that can be determined from the dual algebra for (A, Δ), following the Kantor-Koecher-Tits process. The structure of subcoalgebras and coideals of the coalgebra 〈L(A), Δ_L〉 is characterized. Supported by ISF grant No. RB 6000. Translated fromAlgebra i Logika, Vol. 35, No. 2, pp. 173–189, March–April, 1996.     

A homomorphism of the group GL<Subscript>2</Subscript>(<Emphasis Type="Italic">R</Emphasis>)

We consider homomorphisms of the group GL₂ over any associative ring R with invertible elements 2 and 3. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 3, pp. 95–108, 2005.     

Embeddability and epimorphism relations in undecidable locally finite varieties

Some of the results obtained by A. Pinus for congruence distributive varieties are generalized to the case of congruence-nodular varieties. Most attention is paid to congruence-modular varieties containing a subdirectly indecomposable algebra with non-Abelian monolith. Supported by RFFR grant No. 93-011-1520. Translated fromAlgebra i Logika, Vol. 35, No. 1, pp. 79–87, January–February, 1996.     

Constructibility of Boolean algebras of elementary characteristic (1,0,1)

We deal with problems of finding a criterion of being strongly constructivizable for Boolean algebras. An example of a constructive but not strongly constructivizable Boolean algebra of characteristic (1, 0, 1) with a decidable set of atoms is constructed, and the construction is then generalized to the case of an arbitrary characteristic (k+1,0,1). Supported by the RFFR grant No. 96-01-01525. Translated fromAlgebra i Logika, Vol. 37, No. 5, pp. 499–521, September–October, 1998.