Free subalgebras of the skew polynomial rings k[x,y][t;\sigma] and k[x,x^{-1},y,y^{-1}][t;\sigma]

TL;DR

This paper investigates conditions under which subalgebras generated by specific elements in skew polynomial rings are free, focusing on automorphisms of polynomial and Laurent polynomial rings and their spectral properties.

Contribution

It characterizes when subalgebras generated by certain elements are free in skew polynomial rings based on automorphism properties and spectral radius criteria.

Findings

Subalgebra generated by xt and yt is free iff   is not conjugate to an elementary automorphism.

In Laurent polynomial rings, the spectral radius > 2 guarantees a free subalgebra.

Contains free subalgebras iff spectral radius of the associated matrix > 1.

Abstract

Let \sigma be an automorphism of a commutative k-algebra R. The skew polynomial ring R[t;\sigma] is generated by R and an indeterminate t subject to the relations ta=\sigma(a)t for all a in R. For certain R and appropriate \sigma there are elements a and b in R such that the subalgebra of R[t;\sigma] generated by at and bt is a free algebra. If \sigma is an automorphism of the polynomial ring k[x,y], then the subalgebra of k[x,y][t;\sigma] generated by xt and yt is free if and only if \sigma is not conjugate to an elementary automorphism. If \sigma is an automorphism of k[x,x^{-1},y,y^{- 1}] of the form \sigma(x)=x^ay^b and \sigma(y)=x^cy^d, then the subalgebra of k[x,x^{-1},y,y^{- 1}][t;\sigma] generated by xt and yt is free if the spectral radius of the 2x2 matrix {{a b} \\ {c d}} is >2; indeed, k[x,x^{-1},y,y^{- 1}][t;\sigma] contains a free subalgebra if and only if the spectral radius of 2x2 matrix {{a b} \\ {c d}} is >1.