Odd automorphisms of two generated braided free associative algebras

Abstract

It is proved that an endomorphism ϕ of an braided free associative algebra in two generators over an
arbitrary field k with an involutive diagonal braiding τ = (−1, −1, −1, −1) given by the rule ϕ(x1) = x1, ϕ(x2) =
αx2 +βxm
1
, where α, β ∈ k, m is an odd number, is an odd automorphism. It is also proved that the linear endomorphism
ψ of this algebra is an automorphism if and only if ψ 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 τ = (−1, −1, −1, −1)
coincides with the group of odd automorphisms of this algebra.