Нечетные автоморфизмы сплетенных свободных ассоциативных алгебр от двух порождающих

Abstract

It is proved that an endomorphism $\varphi$ of an braided free associative algebra in two generators over an arbitrary field $k$ with an involutive diagonal braiding $\tau = (- 1, -1, -1, -1)$ given by the rule $\varphi (x_1) = x_1, \, \varphi (x_2) = \alpha x_2 + \beta x^m_1,$ where $\alpha, \, \beta \in k, \, m $ is an odd number, is an odd automorphism. It is also proved that the linear endomorphism $\psi$ of this algebra is an automorphism if and only if $\psi$ is affine. It is shown that the group of all automorphisms of braided free associative algebra in two variables over an arbitrary field $ k $ with an involutive diagonal braiding $ \tau = (- 1, -1, -1, -1) $ coincides with the group of odd automorphisms of this algebra.