Automorphisms of the skew field of rational quantum functions (Q2736050)

Let \(D\) be a skew field with automorphisms \(\alpha_1,\dots,\alpha_n\), where \(n\geq 2\). Given \(q_{ij}\in D^\times\) such that \(q_{ii}=q_{ij}q_{ji}=Q_{ijr}Q_{jri}Q_{rij}=1\), \(\alpha_i(\alpha_j(d))=q_{ij}\alpha_j(\alpha_i(d))q_{ji}\), where \(Q_{ijr}=q_{ij}\alpha_j(q_{ir})\) and \(d\in D\). Define a matrix \(Q=(q_{ij})\) and a row vector \(\alpha=(\alpha_1,\dots,\alpha_n)\) and define \(\Lambda\) as the ring of quantum polynomials on \(X_1,\dots,X_n\) with commutation rules \(X_id=\alpha_i(d)X_i\), \(X_iX_j=q_{ij}X_jX_i\) (\(i\leq i,j\leq n\)). It is an Ore domain and so has a Laurent ring (localization by \(X_1,\dots,X_n\)) and a skew field of fractions \(F\). For the case \(n=2\) the automorphisms of the skew field \(k_q(X,Y)\) were described by \textit{V. A. Artamonov} and \textit{P. M. Cohn} [J. Math. Sci., New York 93, No. 6, 824-829 (1999; Zbl 0928.16029)].NEWLINENEWLINENEWLINEThe author's aim is to describe, for \(n\geq 3\), the group \(\Aut F\) of all automorphisms of \(F\) that are trivial on \(D\). Define a valuation \(\|\cdot\|\) on \(\Lambda\) by putting \(\|f\|=(l_1,\dots,l_n)\) if \(f\) is non-zero with the lowest monomial \(X^{l_1}_1\cdots X^{l_n}_n\) with the lexicographic order on \(\mathbb{Z}^n\). The author proves that for any \(\gamma\in\Aut F\) there exist \(\gamma_1,\dots,\gamma_n\in D^\times\), \(u_1,\dots,u_n\in F\) and \(\varepsilon=\pm 1\) such that \(\gamma(X_i)=\gamma_iX^\varepsilon_i+u_i\), where \(\|u_i\|>\varepsilon\|X_i\|\) (\(i=1,\dots,n)\). Moreover, the map \(\theta\) on \(F\) such that \(\theta(X_i)=\gamma_iX^\varepsilon_i\), \(\theta(d)=d\) for \(d\in D\), is an automorphism of \(F\). When \(\varepsilon=1\), the \(\gamma\)'s are central in \(D\), while for \(\varepsilon=-1\) they satisfy a relation with the \(\alpha\)'s and \(\alpha_i(\gamma_i)\gamma^{-1}_i\) lies in the centre of \(D\). The subgroup of automorphisms with \(\varepsilon=1\) is expressed as a semidirect product and other special cases are considered (where the \(\gamma\)'s are central). -- Next the skew field \(\mathcal F\) of formal Laurent series is constructed and is shown to contain a subfield isomorphic to \(F\). It is shown that if \(D\) is a field of characteristic \(0\), the \(\alpha\)'s act trivially on \(D\) and the \(q\)'s satisfy a certain independence condition, then the centralizer of any \(f\in{\mathcal F}\setminus D\) is commutative. In particular, the centre of \(\mathcal F\) is then contained in \(D\).