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arXiv · 1308.1694

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

S. Paul Smith
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Abstract

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

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S. Paul Smith. 2013-08-07. Free subalgebras of the skew polynomial rings k[x,y][t;σ] and k[x,x^{-1},y,y^{-1}][t;σ]. https://arxiv.org/abs/1308.1694

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