Automorphic equivalence problem for free associative algebras of rank two

Let K be the free associative algebra of rank 2 over an algebraically closed constructive field of any characteristic. We present an algorithm which decides whether or not two elements in K are equivalent under an automorphism of K. A modification of our algorithm solves the problem whether or not an element in K is a semiinvariant of a nontrivial automorphism. In particular, it determines whether or not the element has a nontrivial stabilizer in Aut K. An algorithm for equivalence of polynomials under automorphisms of C[x,y] was presented by Wightwick. Another, much simpler algorithm for automorphic equivalence of two polynomials in K[x,y] for any algebraically closed constructive field K was given by Makar-Limanov, Shpilrain, and Yu. In our approach we combine an idea of the latter three authors with an idea from the unpubished thesis of Lane used to describe automorphisms which stabilize elements of K. This also allows us to give a simple proof of the corresponding result for K[x,y] obtained by Makar-Limanov, Shpilrain, and Yu

Coordinates and Automorphisms of Polynomial and Free Associative Algebras of Rank Three

We study z-automorphisms of the polynomial algebra K[x,y,z] and the free associative algebra K over a field K, i.e., automorphisms which fix the variable z. We survey some recent results on such automorphisms and on the corresponding coordinates. For K we include also results about the structure of the z-tame automorphisms and algorithms which recognize z-tame automorphisms and z-tame coordinates

Tame Automorphisms Fixing a Variable of Free Associative Algebras of Rank Three

We study automorphisms of the free associative algebra K over a field K which fix the variable z. We describe the structure of the group of z-tame automorphisms and derive algorithms which recognize z-tame automorphisms and z-tame coordinates

Cusp Eigenforms and the Hall Algebra of an Elliptic Curve

We give an explicit construction of the cusp eigenforms on an elliptic curve defined over a finite field using the theory of Hall algebras and the Langlands correspondence for function fields and \GL_n. As a consequence we obtain a description of the Hall algebra of an elliptic curve as an infinite tensor product of simpler algebras. We prove that all these algebras are specializations of a universal spherical Hall algebra (as defined and studied in \cite{BS} and \cite{SV1})

The strong Anick conjecture is true

Recently Umirbaev has proved the long-standing Anick conjecture, that is, there exist wild automorphisms of the free associative algebra K over a field K of characteristic 0. In particular, the well-known Anick automorphism is wild. In this article we obtain a stronger result (the Strong Anick Conjecture that implies the Anick Conjecture). Namely, we prove that there exist wild coordinates of K. In particular, the two nontrivial coordinates in the Anick automorphism are both wild. We establish a similar result for several large classes of automorphisms of K. We also find a large new class of wild automorphisms of K which is not covered by the results of Umirbaev. Finally, we study the lifting problem for automorphisms and coordinates of polynomial algebras, free metabelian algebras and free associative algebras and obtain some interesting new results.Comment: 25 pages, corrected typos and acknowledgement