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


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


Zbl 0694.17007
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