The Chevalley and Costant theorems for Mal’tsev algebras

Centers of universal envelopes for Mal’tsev algebras are explored. It is proved that the center of the universal envelope for a finite-dimensional semisimple Mal’tsev algebra over a field of characteristic 0 is a ring of polynomials in a finite number of variables equal to the dimension of its Cartan subalgebra, and that universal enveloping algebra is a free module over its center. Centers of universal enveloping algebras are computed for some Mal’tsev algebras of small dimensions. Supported by FAPESP grant No. 04/08537-4 and by SO RAN grant No. 1.9. Supported by FAPESP grant Nos. 05/60142-7, 05/60337-2 and by CNPq grant No. 304991/2006-6. __________ Translated from Algebra i Logika, Vol. 46, No. 5, pp. 560–584, September–October, 2007.     

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.     

Isomorphically rigid algebras

Various classes of non-associative algebras possessing the property of being rigid under abstract isomorphisms are studied. Supported by RFBR grant No. 06-01-00159a. __________ Translated from Algebra i Logika, Vol. 46, No. 4, pp. 483–502, July–August, 2007.     

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 Filtration of Wick Algebra and Its Application to Quantum SDEs

A family of closed snbalgebras, indexed by R(the set of real numbers), of the Wick algebra is constructed. Fundamental properties of tile family are shown including the increasing property and the right-continuity. The notion of adaptedness to the family is defined for quantum stochastic processes in terms of generalized operators. The existence and uniqueness of solutions adapted to the family is established for quantum stochastic differential equations in terms of generalized operators.     

Generalized entropy in expanded semigroups and in algebras with neutral element

The complex product of (non-empty) subalgebras of a given algebra from a variety \(\mathcal{V}\) is again a subalgebra if and only if the variety \(\mathcal{V}\) has the so-called generalized entropic property. This paper is devoted to algebras with a neutral element or with a semigroup operation. We investigate relationships between the generalized entropic property and the commutativity of the fundamental operations of the algebra. In particular, we characterize the algebras with a neutral element that have the generalized entropic property. Furthermore, we show that, similarly as for n-monoids and n-groups, for inverse semigroups, the generalized entropic property is equivalent to commutativity.     

Subadditivity Inequalities in von Neumann Algebras and Characterization of Tracial Functionals

We examine under which assumptions on a positive normal functional φ on a von Neumann algebra, and a Borel measurable function f: R ⁺ → R with f(0) = 0 the subadditivity inequality φ (f(A+B)) ≤ φ(f(A))+φ (f (B)) holds true for all positive operators A, B in . A corresponding characterization of tracial functionals among positive normal functionals on a von Neumann algebra is presented. O.E. Tikhonov - Supported by the Russian Foundation for Basic Research (grant no. 01-01-00129) and the scientific program Universities of Russia – Basic Research (grant no. UR.04.01.061).     

The Beth property and interpolation in lattice-based algebras and logics

We deal with logics based on lattices with an additional unary operation. Interrelations of different versions of interpolation, the Beth property, and amalgamation, as they bear on modal logics and varieties of modal algebras, superintuitionistic logics and varieties of Heyting algebras, positive logics and varieties of implicative lattices, have been studied in many works. Sometimes these relations can and sometimes cannot be extended to the logics without implication considered in the paper. Supported by INTAS (grant No. 04-77-7080) and by RFBR (grant No. 06-01-00358). Supported by INTAS grant No. 04-77-7080. __________ Translated from Algebra i Logika, Vol. 47, No. 3, pp. 307–334, May–June, 2008.     

Borel Subalgebras Redux with Examples from Algebraic and Quantum Groups

Both building upon and revising previous literature, this paper formulates the general notion of a Borel subalgebra B of a quasi-hereditary algebra A. We present various general constructions of Borel subalgebras, establish a triangular factorization of A, and relate the concept to graded Kazhdan–Lusztig theories in the sense of Cline et al. (Tôhoku Math. J. 45 (1993), 511–534). Various interesting types of Borel subalgebras arise naturally in different contexts. For example, `excellent" Borel subalgebras come about by abstracting the theory of Schubert varieties. Numerous examples from algebraic groups, q-Schur algebras, and quantum groups are considered in detail.     

Finite-dimensional Jordan algebras admitting the structure of a Jordan bialgebra

We define the concepts of a triangular and a quasitriangular Jordan bialgebras. It is proved that a finite-dimensional Jordan algebra J over an algebraically closed field Φ admits the structure of a quasitriangular Jordan bialgebra with nonzero comultiplication, provided that J is not a direct sum of fields, algebras H(Φ₂) and H(Φ₃), null extensions of Φ, and of algebras with zero multiplication. Supported by RFFR grant No. 98-01-01142. Translated fromAlgebra i Logika, Vol. 38, No. 1, pp. 40–67, January–February, 1999.     

Dominions of universal algebras and projective properties

Let A be a universal algebra and H its subalgebra. The dominion of H in A (in a class {ie304-01}) is the set of all elements a ∈ A such that every pair of homomorphisms f, g: A → ∈ {ie304-02} satisfies the following: if f and g coincide on H, then f(a) = g(a). A dominion is a closure operator on a set of subalgebras of a given algebra. The present account treats of closed subalgebras, i.e., those subalgebras H whose dominions coincide with H. We introduce projective properties of quasivarieties which are similar to the projective Beth properties dealt with in nonclassical logics, and provide a characterization of closed algebras in the language of the new properties. It is also proved that in every quasivariety of torsion-free nilpotent groups of class at most 2, a divisible Abelian subgroup H is closed in each group 〈H, a〉 generated by one element modulo H. Translated from Algebra i Logika, Vol. 47, No. 5, pp. 541–557, September–October, 2008.     

Gelfand–Kirillov dimension of generalized Weyl algebras (In memory of Guenter Rudolf Krause (1941–2015))

Given a generalized Weyl algebra A of degree 1 with the base algebra D, we prove that the difference of the Gelfand–Kirillov dimension of A and that of D could be any positive integer or infinity. Under mild conditions, this difference is exactly 1. As applications, we calculate the Gelfand–Kirillov dimensions of various algebras of interest, including the (quantized) Weyl algebras, ambiskew polynomial rings, noetherian (generalized) down-up algebras, iterated Ore extensions, quantum Heisenberg algebras, universal enveloping algebras of Lie algebras, quantum GWAs, etc.     

The Maximal Graded Left Quotient Algebra of a Graded Algebra1)

We construct the maximal graded left quotient algebra of every graded algebra A without homogeneous total right zero divisors as the direct limit of graded homomorphisms （of left A-modules） from graded dense left ideals of A into a graded left quotient algebra of A. In the case of a superalgebra, and with some extra hypothesis, we prove that the component in the neutral element of the group of the maximal graded left quotient algebra coincides with the maximal left quotient algebra of the component in the neutral element of the group of the superalgebra.     

A weak form of interpolation in equational logic

The notions of a weak interpolation property and of weak amalgamation are introduced. It is proved that in varieties with the congruence extension property, the weak interpolation property is equivalent to the weak amalgamation property. In turn, weak amalgamability of a variety is equivalent to amalgamability of a class of finitely generated simple algebras in this variety. Supported by RFBR (grant Nos. 06-01-00358 and 05-01-04003-NNIOa) and by INTAS (grant No. 04-77-7080). __________ Translated from Algebra i Logika, Vol. 47, No. 1, pp. 94–107, January–February, 2008.     

Globalizing local actions of coalgebras

It is proved that any continuous weakly invertible local measuring of an algebra with a coalgebra extends to the global measuring of a quotient algebra. Also, any twisted measuring with an invertible cocycle is shown to be weakly invertible, and sufficient conditions are specified for pointed and connected algebras to have continuous measurings. Supported by RFFR grant No. 95-01-01356a and by ISF grants Nos. RPS 000 and RPS 300. Translated fromAlgebra i Logika, Vol. 35, No. 2, pp. 190–213, March–April, 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.     

Test elements and the retract theorem for monounary algebras

The term “Retract Theorem” has been applied in literature in connection with group theory. In the present paper we prove that the Retract Theorem is valid (i) for each finite structure, and (ii) for each monounary algebra. On the other hand, we show that this theorem fails to be valid, in general, for algebras of the form A=(A, F), where each f ∈ F is unary and card F > 1. This work was supported by grant VEGA 1/0423/03 and by Science and Technology Assistance Agency under the contract No.APVT-20-004104.     

Representation theory of q-rook monoid algebras

We show that, over an arbitrary field, q-rook monoid algebras are iterated inflations of Iwahori-Hecke algebras, and, in particular, are cellular. Furthermore we give an algebra decomposition which shows a q-rook monoid algebra is Morita equivalent to a direct sum of Iwahori-Hecke algebras. We state some of the consequences for the representation theory of q-rook monoid algebras.Supported by EPSRC grant GR/S18151/01     

Composition algebras of the second kind

The concept of a composition algebra of the second kind is introduced. We prove that such algebras are non-degenerate monocomposition algebras without unity. A big number of these algebras in any finite dimension are constructed, as well as two algebras in a countable dimension. The constructed algebras each contains a non-isotropic idempotent e² = e. We describe all orthogonally non-isomorphic composition algebras of the second kind in the following forms: (1) a two-dimensional algebra (which has turned out to be unique); (2) three-dimensional algebras in the constructed series. For every algebra A, the group Ortaut A of orthogonal automorphisms is specified. __________ Translated from Algebra i Logika, Vol. 46, No. 4, pp. 428–447, July–August, 2007.     

Hyperinvariant subspaces for some <Emphasis Type="Italic">B</Emphasis>–circular operators

We show that if A is a Hilbert–space operator, then the set of all projections onto hyperinvariant subspaces of A, which is contained in the von Neumann algebra υN(A) that is generated by A, is independent of the representation of υ N(A), thought of as an abstract W^*–algebra. We modify a technique of Foias, Ko, Jung and Pearcy to get a method for finding nontrivial hyperinvariant subspaces of certain operators in finite von Neumann algebras. We introduce the B–circular operators as a special case of Speicher's B–Gaussian operators in free probability theory, and we prove several results about a B–circular operator z, including formulas for the B–valued Cauchy– and R–transforms of z^*z. We show that a large class of L^∞([0,1])–circular operators in finite von Neumann algebras have nontrivial hyperinvariant subspaces, and that another large class of them can be embedded in the free group factor L(F₃). These results generalize some of what is known about the quasinilpotent DT–operator. Supported in part by NSF Grant DMS-0300336. with an Appendix by Gabriel Tucci