Quantum integrability of the generalized Euler's top with symmetry (Q2711769)

This paper deals with the quantum systems that are higher-rank generalization of the standard \(so(3)\) Euler top. Let \(\mathfrak g\) be a Lie algebra, let \(M_i\) be coordinate functions on the dual space \(\mathfrak g^*\), and let \((P(\mathfrak g^*),\{\;,\;\})\) be the Poissonian manifold of functions on \(\mathfrak g^*\) with the Lie-Poisson bracket \(\{\;,\;\}\). Quantization is the map \(M_ i \mapsto \widehat M_i\) from the set of coordinates into the set of Hermitian operators in some Hilbert space \(\mathcal{H}\), so that \(\{\widehat{M_i,M_j}\}=-i\hbar[\widehat M_i,\widehat M_j]\). The author searches for commutative quantum counterparts of Mishchenko-Fomenko algebra of involutive functions on \((P(\mathfrak g^*),\{\;,\;\})\) constructed by the ``shift of the argument'' procedure [see \textit{A. S. Mishchenko} and \textit{A. T. Fomenko}, Math. USSR, Izv. 12, 371-389 (1978); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 42, 396-415 (1978; Zbl 0383.58006)]. He shows that for Lie algebra \(\mathfrak g\) of the type \(gl(n), so(n)\) or \(sp(n)\) the quantum Euler's top with the inertia tensor \(A\in \mathfrak g\) is integrable if the centralizer of \(A\) contain Lie subalgebra of the type \(gl(n-2), so(n-2)\) or \(sp(n-1)\) respectively. On the other hand, one obtains the integrable quantum Euler top with arbitrary inertia tensor in the case where the associated Euler-Arnold equations are restricted to the degenerated coadjoint orbit whose stabilizer \(K\) contains Lie subgroups of the type \(Gl(n-2), SO(n -2)\) or \(Sp(n-1)\) respectively.NEWLINENEWLINEFor the entire collection see [Zbl 0937.00046].