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Gel’fand-Najmark-Segal states and the Weyl group for the quantum group SU(n). (English. Russian original) Zbl 0712.17017

Funct. Anal. Appl. 24, No. 3, 253-255 (1990); translation from Funkts. Anal. Prilozh. 24, No. 3, 92-93 (1990).
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.
Reviewer: D.Leites

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
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