On the automorphism groups of relatively free groups of infinite rank: a survey

The paper is intended to be a survey on some topics within the framework of automorphisms of a relatively free groups of infinite rank. We discuss such properties as tameness, primitivity, small index, Bergman property, and so on

A polynomial algorithm for the braid double shielded public key cryptosystems

We propose new provable practical deterministic polynomial time algorithm of cryptographic analysis for the braid Wang, Xu, Li, Lin and Wang «Double shielded public key cryptosystems», where the authors recommended the Artin braid groups Bn as platforms for proposed protocols. We show that a linear decomposition attack based on the decomposition method introduced by the author works for the image of braids under the Lawrence-Krammer representation by finding the exchanging keys in the both two main protocols by Wang et. al. These keys can be effectively computed in their original form too. Thus the protocols propоsed by Wang et. al. are vulnerable

О разрешимости коммутаторных уравнений в алгебрах Ли

Доказано, что любое коммутаторное уравнение разрешимо над алгеброй Ли Гейзенберга над произвольным полем в большей алгебре Ли верхних нильтреугольных матриц над этим же полем. Показано, что любой член нижнего центрального ряда кольца (алгебры) Ли верхних нильтреугольных матриц является образом коммутаторной функции от одной переменной, определенной на этом кольце (алгебре)