Quantum function algebras as quantum enveloping algebras

Abstract

Inspired by a result in [Ga], we locate two $k[q,q^{-1}]$-integer forms of $F_q[SL(n+1)]$, along with a presentation by generators and relations, and prove that for $q=1$ they specialize to $U(\mathfrak{h})$, where $\mathfrak{h}$ is the Lie bialgebra of the Poisson Lie group $H$ dual of $SL(n+1)$; moreover, we explain the relation with [loc. cit.]. In sight of this, we prove two PBW-like theorems for $F_q[SL(n+1)]$, both related to the classical PBW theorem for $U(\mathfrak{h})$.

27 pages, AMS-TeX C, Version 3.0 - Author's file of the final version, as it appears in the journal printed version, BUT for a formula in Subsec. 3.5 and one in Subsec. 5.2 - six lines after (5.1) - that in this very pre(post)print have been corrected

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• Quantization of Poisson Groups
• The global quantum duality principle: a survey through examples
• A quantum homogeneous space of nilpotent matrices
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• F_q[M_n], F_q[GL_n] and F_q[SL_n] as quantized hyperalgebras