On the Gelfand-Kirillov conjecture for quantum algebras

Abstract

Let q q be a complex not a root of unity and g \mathfrak {g} be a semi-simple Lie C \mathbb {C} -algebra. Let U q ( g ) U_{q}(\mathfrak {g}) be the quantized enveloping algebra of Drinfeld and Jimbo, U q ( n − ) ⊗ U 0 ⊗ U q ( n ) U_{q}(\mathfrak {n}^-)\otimes U^{0}\otimes U_{q}(\mathfrak {n}) be its triangular decomposition, and C q [ G ] \mathbb {C}_{q}[G] the associated quantum group. We describe explicitly Fract ⁡ U q ( n ) \operatorname {Fract} U_{q}(\mathfrak {n}) and Fract ⁡ C q [ G ] \operatorname {Fract}\mathbb {C}_{q}[G] as a quantum Weyl field. We use for this a quantum analogue of the Taylor lemma.