Open AccessOdd automorphisms of two generated braided free associative algebras
Author(s) -
Riza Mutalip,
Altyngul Naurazbekova
Publication year - 2020
Publication title -
l.n. gumilev atyndaġy euraziâ u̇lttyk̦ universitetìnìn̦ habaršysy. matematika, informatika, mehanika seriâsy
Language(s) - English
Resource type - Journals
eISSN - 2663-1326
pISSN - 2616-7182
DOI - 10.32523/2616-7182/2020-131-2-42-50
Subject(s) - automorphism , endomorphism , mathematics , associative property , associative algebra , diagonal , pure mathematics , field (mathematics) , group (periodic table) , division algebra , algebra over a field , free group , combinatorics , filtered algebra , physics , geometry , quantum mechanics
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.