Torus actions on free associative algebras, lifting and Białynicki-Birula type theorems
Alexei Belov-Kanel, Andrey Elishev, Farrokh Razavinia, Louis Rowen, Jie-Tai Yu, Wenchao Zhang
TL;DR
The paper extends the classical torus action linearization problem to the free associative algebra setting, proving that any effective regular action of the maximal torus $\mathbb{T}_n$ on the free algebra $F_n$ is linearizable and establishing linearization for positive-root torus actions via two complementary approaches (generic-matrix reduction and Yagzhev’s rational-invertibility framework). It analyzes the $K^*$ action in the $n=2$ case, discusses conjectural extensions to higher ranks, and links linearization to deep conjectures such as the Jacobian and associative cancellation problems. The work also develops an Asanuma-type framework to construct non-linearizable torus actions in positive characteristic and explores consequences for non-rectifiable epimorphisms and the cancellation phenomenon in the associative setting. Together, these results deepen the understanding of noncommutative torus actions, illuminate connections to cancellation and Jacobian-type questions, and propose pathways for further generalizations, including Poisson and broader noncommutative contexts.
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.