Automorphisms of certain completions of quantum Weyl skewfields
Let \(k\) be a field and \(\sigma\) the \(k\)-automorphism of \(k((Y))\) defined by \(\sigma(Y) = qY\), where \(q\) is an element of \(k^*\). The purpose of this article is to describe the group \(\text{Aut}_k(L_q)\) for the extension \(L_q = k((Y)) ((X;\sigma))\) of the quantum Weyl skewfield \(D^q_1 = k(Y) (X;\sigma)\), when \(q\) is not a root of 1. An automorphism \(\theta\) of \(L_q\) is said to be triangular if \(\theta(Y) = \alpha Y\) and \(\theta(X) = f(Y) X\) for some \(\alpha\) in \(k^*\) and \(f(Y) \neq 0\) in \(k((Y))\). Denoting by \(S\) the subgroup of the triangular automorphisms of \(L_q\), the main theorem establishes that \(\text{Aut}_k(L_q)/\text{Int} (L_q)\) is isomorphic to \(S/(\text{Int} (L_q) \cap S)\), which is a semi-direct product of \((k^*/\langle q\rangle)\) by \((k^*/\langle q \rangle \times \mathbb{Z})\). The motivations of this result are detailed in the first part of the paper, devoted to the two-dimensional quantum Cremona transformations. Its proof is based on a general result concerning the continuity of automorphisms in skewfields of Laurent series, which also holds for complete extensions of the classical Weyl skewfield \(D_1\).
two-dimensional quantum Cremona transformations
scientific article; zbMATH DE number 869939