2013/08/07 by S. Paul Smith, Smith, S. Paul
Mathematics · #Advanced Differential Equations and Dynamical Systems #Advanced Topics in Algebra #Algebraic structures and combinatorial models #math.RA #msc:16S10 #msc:16S35 #msc:16W20
paper · pdf · doi:10.48550/arxiv.1308.1694
14 pages
arxiv created 2013/08/07 · arxiv updated 2013/08/09
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)=xayb and σ(y)=xcyd, 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.