Group-like Elements in Quantum Groups and Feigin’s Conjecture

Let A be an arbitrary symmetrizable Cartan matrix of rank r, and n = n+ be the standard maximal nilpotent subalgebra in the Kac-Moody algebra associated with A (thus, n is generated by E1, . . . , Er subject to the Serre relations). Let Ûq(n) be the completion (with respect to the natural grading) of the quantized enveloping algebra of n. For a sequence i = (i1, . . . , im) with 1 ≤ ik ≤ r, let Pi be a skew polynomial algebra generated by t1, . . . , tm subject to the relations tltk = q Cik,il tktl (1 ≤ k < l ≤ m) where C = (Cij) = (diaij) is the symmetric matrix corresponding to A. We construct a group-like element ei ∈ Pi ⊗ Ûq(n). This element gives rise to the evaluation homomorphism ψi : Cq[N ] → Pi given by ψi(x) = x(ei), where Cq[N ] = Uq(n) 0 is the restricted dual of Uq(n). Under a well-known isomorphism of algebras Cq[N ] and Uq(n), the map ψi identifies with Feigin’s homomorphism Φ(i) : Uq(n) → Pi. We prove that the image of ψi generates the skew-field of fractions F(Pi) if and only if i is a reduced expression of some element w in the Weyl group W ; furthermore, in the latter case, Ker ψi depends only on w (so we denote Iw := Ker ψi). This result generalizes the results in [5], [6] to the case of Kac-Moody algebras. We also construct an element Rw ∈ ( Cq[N ]/Iw )⊗ Ûq(n) which specializes to ei under the embedding Cq[N ]/Iw →֒ Pi. The elements Rw are closely related to the quaziR-matrix studied by G. Lusztig in [8]. If i, i are reduced expressions of the same element w ∈ W , we have a natural isomorphism R ′ i : F(Pi) → F(Pi′) such that (R ′ i ⊗ id)(ei) = ei′ . This leads to identities between quantum exponentials. The maps R ′ i are q-deformations of Lusztig’s transition maps [8]. The existence of the maps R ′ i leads to a surprising combinatorial corollary about skew-symmetric matrices associated with reduced expressions (cf. [12]).