Gel'fand-Najmark-Segal states and the Weyl group for the quantum group SU(n) (Q923712)

The Gel'fand-Najmark-Segal states, mentioned in the paper only once: in the title, are, however, not only for publicity's sake. Their ``quantum'' analogues for SU(n) are listed in Theorem 1 (as the reader might guess). A quasiclassical interpretation of Theorem 1 gives an explicit description of symplectic leaves of SU(n). The algebra of regular functions \({\mathbb{C}}[SL_ n({\mathbb{C}})]_ h\) on the quantum \(SL_ n({\mathbb{C}})\) is defined and, in \({\mathbb{C}}[SL_ n({\mathbb{C}})]_ h^*\) quantum analogues, \(\bar s_ i\), of the generators of the Weyl group of \(SL_ n({\mathbb{C}})\) are explicitly realised. Their Ad-action on the generators of \(U_ h(sl(n))\) is explicitly written; it identifies \(Ad_{\bar s_ i}\) with automorphisms \(T_ i'\) of \(U_ h(sl(n))\) introduced by G. Lusztig.