Commutative subalgebras of the ring of quantum polynomials and of the skew field of quantum Laurent series (Q2759315)

The author considers the quantum algebra \(\Lambda\) in \(n\) variables over a field \(k\) and its completion \(\mathcal F\), the skew field of quantum Laurent series. The lowest degree of any terms occurring defines a valuation \(\|.\|\) on \(\mathcal F\) with values in \(\mathbb{Z}^n\) and the author proves that if all elements commuting with \(f\in{\mathcal F}\) have as value a multiple of \(\|h\|\), then the centralizer of \(f\) is \(k( (h))\). This generalizes the case \(n=2\), \(\text{char }k=0\) proved by \textit{V. A. Artamonov} and \textit{P. M. Cohn} [J. Math. Sci., New York 93, No. 6, 824-829 (1999; Zbl 0928.16029)]. She also shows that in the Lie algebra derived from \(\mathcal F\) any finite-dimensional subalgebra is Abelian. Her next task is to examine pairs of elements of \(\Lambda\) and she proves that any pair of commuting elements are algebraically dependent over \(k\); it follows that any commutative subalgebra of \(\Lambda\) is isomorphic to a subalgebra of \(k( (h))\) for some \(h\in\Lambda\) and its transcendence degree over \(k\) is at most one. Finally, an example shows that for two commuting elements \(f\) and \(g\) the algebra \(k[f,g]\) need not be isomorphic to \(k[x]\).