[Algebra and Analysis Seminar]
Ivan Shestakov (University of São Paulo and Sobolev Institute of Mathematics, Novosibirsk)
Tame and wild automorphisms of polynomials and free algebras
Abstract: An automorphism f of a free algebra FV[x1, . . . , xn] from a class V over a field F is called elementary if it has a form f(x1,...,xi,...,xn)=(x1,...,a xi+ f,...,xn), where a is a nonzero scalar and the polynomial f does not contain xi.
An automorphism is called tame if it can be represented as a composition of elementary automorphisms. It is known that all automorphisms of the algebra of polynomials and of free associative algebra in two variables are tame, whereas in the case of three variables in both cases there exist wild automorphisms.
In our talk we will discuss known results and open problems on tame and wild automorphisms in different classes of algebras.