Noncommutative Bia lynicki–Birula Theorem

Abstract

The study of algebraic group actions on varieties and coordinate algebras is a major area
of research in algebraic geometry and ring theory. The subject has its connections with the
theory of polynomial mappings, tame and wild automorphisms, the Jacobian conjecture of
O.-H. Keller, infinite-dimensional varieties according to Shafarevich, the cancellation problem
(together with various cancellation-type problems), the theory of locally nilpotent derivations,
among other topics. One of the central problems in the theory of algebraic group actions has
been the linearization problem, formulated and studied in the work of T. Kambayashi and
P. Russell, which states that any algebraic torus action on an affine space is always linear with
respect to some coordinate system. The linearization conjecture was inspired by the classical
and well known result of A. Bia lynicki–Birula, which states that every effective regular torus
action of maximal dimension on the affine space over algebaically closed field is linearizable.
Although the linearization conjecture has turned out negative in its full generality, according
to, among other results, the positive-characteristic counterexamples of T. Asanuma, the
Bia lynicki–Birula has remained an important milestone of the theory thanks to its connection to
the theory of polynomial automorphisms. Recent progress in the latter area has stimulated the
search for various noncommutative analogues of the Bia lynicki–Birula theorem. In this paper, we
give the proof of the linearization theorem for effective maximal torus actions by automorphisms
of the free associative algebra, which is the free analogue of the Bia lynicki–Birula theorem.