Torus actions on free associative algebras, lifting and Bia{\l}ynicki-Birula type theorems

TL;DR

This paper extends classical linearity theorems of torus actions to free associative algebras, introduces new linearity results for specific actions, and explores non-linearizable actions with applications to the Associative Cancellation Conjecture.

Contribution

It proves the free algebra analog of Bialynicki-Birula's theorem, formulates linearity theorems for certain actions, and constructs non-linearizable actions in the associative algebra setting.

Findings

Proved the linearity of maximal torus actions on free algebras.

Developed a framework for constructing non-linearizable torus actions.

Demonstrated the existence of non-isomorphic algebras with isomorphic free products with polynomial rings.

Abstract

We examine the problem of the linearity of an algebraic torus action in the associative setting. We prove the free algebra analog of a classical theorem of BialynickiBirula, which establishes linearity of maximal torus action. Additionally, we formulate and prove linearity theorems for specific classes of regular actions, and provide a framework for constructing non-linearizable actions, analogous to the work of Asanuma. This framework has applications in the study of the Associative Cancellation Conjecture. Furthermore, we show the existence of two non-isomorphic algebras, whose free products with a polynomial ring are isomorphic.