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Irreducible representations of the function algebra on the quantum group \(\text{SU}(n)\), and Schubert cells. (English. Russian original) Zbl 0698.22015

Sov. Math., Dokl. 40, No. 1, 34-38 (1990); translation from Dokl. Akad. Nauk SSSR 307, No. 1, 41-45 (1989).
Let \(G\) be a simple complex Lie group, \(\mathfrak{g}\) its Lie algebra, \(U(\mathfrak{g})_ h\) the quantum enveloping algebra of \(\mathfrak{g}\). The matrix elements of finite dimensional unitarizable representations of \(U(\mathfrak{g})_ h\) for which \(\rho (E_ i)\) is a positive operator form the Hopf algebra \(C[G]_ h\). It is called the algebra of regular functions on the quantum group. In this paper, a Hopf *-algebra is obtained from \(C[G]_ h\) and denoted by \(C[K]_ h\), where \(K\) is the maximal compact subgroup. If \(G=\text{SL}_ n(\mathbb{C})\) and \(K=\text{SU}(n)\), it is proved that \(V_ 1\simeq V_ 2\) if and only if \(\Lambda_ 1=\Lambda_ 2\), where \(V_ i\), \(i=1,2\), are simple \(C[K]_ h\)-modules, corresponding to the Schubert cell \(X_ w\) and having the highest weights \(\Lambda_ i\). A condition on the weight \(\Lambda\) for which a simple \(C[K]_ h\)-module with highest weight \(\Lambda\) corresponding to the cell \(X_ w\) exists is given.
Reviewer: Li Wanglai

MSC:

22E60 Lie algebras of Lie groups
20G42 Quantum groups (quantized function algebras) and their representations
16T05 Hopf algebras and their applications
17B37 Quantum groups (quantized enveloping algebras) and related deformations
17B62 Lie bialgebras; Lie coalgebras
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