THE p-ZASSENHAUS FILTRATION OF A FREE PROFINITE GROUP AND SHUFFLE RELATIONS

For a prime number p and a free profinite group S on the basis X, let $S_{\left (n,p\right )}$ , $n=1,2,\dotsc ,$ be the p-Zassenhaus filtration of S. For $p>n$ , we give a word-combinatorial description of the cohomology group $H^2\left (S/S_{\left (n,p\right )},\mathbb {Z}/p\right )$ in terms of the shuffle algebra on X. We give a natural linear basis for this cohomology group, which is constructed by means of unitriangular representations arising from Lyndon words.

On the construction of staggered linear bases

A staggered linear basis is a specific basis for a polynomial ideal generating the ideal as a linear vector space. There are some algorithms in the literature to compute these bases and we show that they do not work properly. In this paper, due to the strong connection between Gröbner bases and staggered linear bases and by applying a signature-based structure, we present a new and correct algorithm to compute staggered linear bases. Since the output of this algorithm also remains a Gröbner basis for its input ideal, we conclude the paper by discussing the performance of this algorithm compared with the newest signature-based algorithm to compute Gröbner bases.

Palindromes in the free metabelian Lie algebras

A palindrome, in general, is a word in a fixed alphabet which is preserved when taken in reverse order. Let [Formula: see text] be the free metabelian Lie algebra over a field of characteristic zero generated by [Formula: see text]. We propose the following definition of palindromes in the setting of Lie algebras: An element [Formula: see text] is called a palindrome if it is preserved under the change of generators; i.e. [Formula: see text]. We give a linear basis and an explicit infinite generating set for the Lie subalgebra of palindromes.