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An analogue of P.B.W. theorem and the universal R-matrix for \(U_ h\mathfrak{sl}(N+1)\). (English) Zbl 0694.17006
For the “quantum” deformation of \(\mathfrak{sl}(n;{\mathbb C})\) the universal R-matrix, the measure of noncocommutativity of \(U_ h(\mathfrak{sl}(n;{\mathbb C}))\), is explicitly calculated. For this the basis of the quantum deformation \(U_ h(\mathfrak{sl}(n;{\mathbb C}))\) of the enveloping algebra – the P.B.W. theorem – is formulated.
Remark. The answer for the R-matrix might only by accident be true for \(h\) such that \(h^ 8=1\): in this case the P.B.W. theorem reads differently [cf. H. Yamane, Publ. Res. Inst. Math. Sci. 25, No. 3, 503–520 (1989; Zbl 0694.17007)].
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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