New example of a variety of lie algebras with fractional exponent

It is shown that in the case of characteristic zero the variety generated by a simple infinitedimensional Lie algebra of Cartan type W ₂ has a fractional exponent.     

An example of Clifford algebras calculations with GiNaC

This is an example of C++ code of Clifford algebra calculations with the GiNaC computer algebra system. This code makes both symbolic and numeric computations. It was used to produce illustrations for paper [14, 12]. Described features of GiNaC are already available at PyGiNaC [3] and due to course should propagate into other software like GNU Octave [7] and gTybalt [18] which use GiNaC library as their back-end. *On leave from Odessa University.     

Equivalent conditions of polynomial growth of a variety of Poisson algebras

Equivalent conditions of the polynomial codimension growth of a variety of Poisson algebras over a field of characteristic zero are presented and it is shown that there are only two varieties of Poisson algebras with almost polynomial growth.     

An example on ordered Banach algebras

Let be a complex unital Banach algebra. We consider the Banach algebra ordered by the algebra cone , and investigate the connection between results on ordered Banach algebras and the right bound of the numerical range of elements in .

     

An independent system of identities for a variety of mono-Leibniz algebras

We introduce the notion of a mono-Leibniz algebra generalizing the concept of a Leibniz algebra. Namely, an algebra A over a field K, charK ≠ 2, is mono-Leibniz if its one-generated subalgebras each is a Leibniz algebra. It is proved that a variety W of mono-Leibniz algebras over an infinite field K is defined by an independent system of identities such as x(xx) = 0 and x[(xx)x] = 0. Examples of mono-Leibniz algebras are given which show that W is not a Schreier variety.     

Simple associative conformal algebras of linear growth

We describe simple finitely generated associative conformal algebras of Gel'fand–Kirillov dimension one.     

Example of a linear algebra variety with the polynomial growth lower than three

In the case of characteristic zero, an example of a linear algebra variety with the growth greater than quadratic, but lower than cubic is constructed.     

An example of a piecewise linear ergodic polymorphism

We consider dynamical systems introduced by Vershik and called polymorphisms. In particular, such systems encompass the class of multivalued mappings of a closed interval onto itself which have an invariant measure. Polymorphisms arise in different areas of mathematics and mechanics, for example, in the problem of the destruction of the adiabatic invariant. We are concerned with the ergodic properties of polymorphisms. The first section deals with the main notions. In Secs. 2 and 3, we consider an example of a three-parameter family of ergodic polymorphisms formed by piecewise linear mappings.     

On growth of varieties of commutative linear algebras

There exists varieties of commutative linear algebras over a field of zero characteristic whose exponent is equal to α for any real α > 1 and the intermediate growth is n^nb {n^{{n^\beta }}} for any real 0 < β < 1.     

Necessary and sufficient conditions for a variety of Leibniz algebras to have polynomial growth

We study the behavior of the codimension sequence of polynomial identities of Leibniz algebras over a field of characteristic 0. We prove that a variety V has polynomial growth if and only if the condition

Weakly representable relation algebras form a variety

We prove that the class of weakly representable relation algebras is closed under homomorphic images, hence it is a variety. As a corollary we classify the subdirectly irreducible algebras in this class. Received April 3, 2007; accepted in final form February 7, 2008.     

Subdirectly irreducible algebras with hyperidentities of the variety of De Morgan algebras

The paper characterizes the class of subdirectly irreducible algebras satisfying hyperidentities of the variety of De Morgan algebras. Such algebras are called subdirectly irreducible De Morgan quasilattices. The suggested characterization is quite close to that of the classical case of subdirectly irreducible DeMorgan algebras.     

Identities defining a certain variety of right-alternative algebras

There are given identical relations defining the variety of algebras, over an arbitrary but fixed field of characteristic zero, all of whose 2-generator subalgebras are of type (–1, 1).Translated from Matematicheskie Zametki, Vol. 20, No. 2, pp. 161–176, August, 1976.