TWISTED CONJUGACY CLASSES IN R. THOMPSON`S GROUP F

In this article, we prove that any automorphism of R. Thompson`s group F has infinitely many twisted conjugacy classes. The result follows from the work of Brin, together with standard facts about R. Thompson`s group F, and elementary properties of the Reidemeister numbers.

Lie properties of crossed products

Let F-sigma(lambda)vertical bar G vertical bar be a crossed product of a group G and the field F. We study the Lie properties of F-sigma(lambda)vertical bar G vertical bar in order to obtain a characterization of those crossed products which are upper (lower) Lie nilpotent and Lie (n, m)-Engel. (C) 2008 Elsevier Inc. All rights reserved.

Crossed products by twisted partial actions and graded algebras

For a twisted partial action e of a group G on an (associative non-necessarily unital) algebra A over a commutative unital ring k, the crossed product A x(Theta) G is proved to be associative. Given a G-graded k-algebra B = circle plus(g is an element of G) B-g with the mild restriction of homogeneous non-degeneracy, a criteria is established for B to be isomorphic to the crossed product B-1 x(Theta) G for some twisted partial action of G on B-1. The equality BgBg-1 B-g = B-g (for all g is an element of G) is one of the ingredients of the criteria, and if it holds and, moreover, B has enough local units, then it is shown that B is stably isomorphic to a crossed product by a twisted partial action of G. (c) 2008 Elsevier Inc. All rights reserved.

Extensions of homogeneous polynomials ON c(0)(l(2)(i))

We show that a 2-homogeneous polynomial on the complex Banach space c(0)(l(2)(i)) is norm attaining if and only if it is finite (i.e, depends only on finite coordinates). As the consequence, we show that there exists a unique norm-preserving extension for norm-attaining 2-homogeneous polynomials on c(0)(l(2)(i)).

Nuclear elements of degree 6 in the free alternative algebra

We construct five new elements of degree 6 in the nucleus of the free alternative algebra. We use the representation theory of the symmetric group to locate the elements. We use the computer algebra system ALBERT and an extension of ALBERT to express the elements in compact form and to show that these new elements are not a consequence of the known clegree-5 elements in the nucleus. We prove that these five new elements and four known elements form a basis for the subspace of nuclear elements of degree 6. Our calculations are done using modular arithmetic to save memory and time. The calculations can be done in characteristic zero or any prime greater than 6, and similar results are expected. We generated the nuclear elements using prime 103. We check our answer using five other primes.

Finite-dimensional non-associative algebras and codimension growth

Let A be a (non-necessarily associative) finite-dimensional algebra over a field of characteristic zero. A quantitative estimate of the polynomial identities satisfied by A is achieved through the study of the asymptotics of the sequence of codimensions of A. It is well known that for such an algebra this sequence is exponentially bounded. Here we capture the exponential rate of growth of the sequence of codimensions for several classes of algebras including simple algebras with a special non-degenerate form, finite-dimensional Jordan or alternative algebras and many more. In all cases such rate of growth is integer and is explicitly related to the dimension of a subalgebra of A. One of the main tools of independent interest is the construction in the free non-associative algebra of multialternating polynomials satisfying special properties. (C) 2010 Elsevier Inc. All rights reserved.

REIDEMEISTER SPECTRUM FOR METABELIAN GROUPS OF THE FORM Q(n) x Z AND Z[1/p](n) x Z, p PRIME

In this paper, we study the Reidemeister spectrum for metabelian groups of the form Q(n) x Z and Z[1/p](n) x Z. Particular attention is given to the R(infinity)-property of a subfamily of these groups.

Description of costandard modules for Schur superalgebra S(2 vertical bar 1) in positive characteristic

We describe the characters of simple modules and composition factors of costandard modules for S(2 vertical bar 1) in positive characteristics and verify a conjecture of La Scala-Zubkov regarding polynomial superinvariants for GL(2 vertical bar 1).

SIGMA THEORY AND TWISTED CONJUGACY CLASSES

Using Sigma theory we show that for large classes of groups G there is a subgroup H of finite index in Aut(G) such that for phi is an element of H the Reidemeister number R(phi) is infinite. This includes all finitely generated nonpolycyclic groups G that fall into one of the following classes: nilpotent-by-abelian groups of type FP(infinity); groups G/G `` of finite Prufer rank; groups G of type FP(2) without free nonabelian subgroups and with nonpolycyclic maximal metabelian quotient; some direct products of groups; or the pure symmetric automorphism group. Using a different argument we show that the result also holds for 1-ended nonabelian nonsurface limit groups. In some cases, such as with the generalized Thompson`s groups F(n,0) and their finite direct products, H = Aut(G).

Nil graded self-similar algebras

In [19], [24] we introduced a family of self-similar nil Lie algebras L over fields of prime characteristic p > 0 whose properties resemble those of Grigorchuk and Gupta-Sidki groups. The Lie algebra L is generated by two derivations v(1) = partial derivative(1) + t(0)(p-1) (partial derivative(2) + t(1)(p-1) (partial derivative(3) + t(2)(p-1) (partial derivative(4) + t(3)(p-1) (partial derivative(5) + t(4)(p-1) (partial derivative(6) + ...))))), v(2) = partial derivative(2) + t(1)(p-1) (partial derivative(3) + t(2)(p-1) (partial derivative(4) + t(3)(p-1) (partial derivative(5) + t(4)(p-1) (partial derivative(6) + ...)))) of the truncated polynomial ring K[t(i), i is an element of N vertical bar t(j)(p) =0, i is an element of N] in countably many variables. The associative algebra A generated by v(1), v(2) is equipped with a natural Z circle plus Z-gradation. In this paper we show that for p, which is not representable as p = m(2) + m + 1, m is an element of Z, the algebra A is graded nil and can be represented as a sum of two locally nilpotent subalgebras. L. Bartholdi [3] andYa. S. Krylyuk [15] proved that for p = m(2) + m + 1 the algebra A is not graded nil. However, we show that the second family of self-similar Lie algebras introduced in [24] and their associative hulls are always Z(p)-graded, graded nil, and are sums of two locally nilpotent subalgebras.

Nonhomogeneous subalgebras of Lie and special Jordan superalgebras

We consider polynomial identities satisfied by nonhomogeneous subalgebras of Lie and special Jordan superalgebras: we ignore the grading and regard the superalgebra as an ordinary algebra. The Lie case has been studied by Volichenko and Baranov: they found identities in degrees 3, 4 and 5 which imply all the identities in degrees <= 6. We simplify their identities in degree 5, and show that there are no new identities in degree 7. The Jordan case has not previously been studied: we find identities in degrees 3, 4, 5 and 6 which imply all the identities in degrees <= 6, and demonstrate the existence of further new identities in degree 7. our proofs depend on computer algebra: we use the representation theory of the symmetric group, the Hermite normal form of an integer matrix, the LLL algorithm for lattice basis reduction, and the Chinese remainder theorem. (C) 2009 Elsevier Inc. All rights reserved.

A Character Formula for the Category (O)over-tilde of Modules for Affine sl(2)

One may construct, for any function on the integers, an irreducible module of level zero for affine sl(2) using the values of the function as structure constants. The modules constructed using exponential-polynomial functions realize the irreducible modules with finite-dimensional weight spaces in the category (O) over tilde of Chari. In this work, an expression for the formal character of such a module is derived using the highest weight theory of truncations of the loop algebra.