Group-like elements in quantum groups, and Feigin's conjecture

Abstract

Let $A$ be an arbitrary symmetrizable Cartan matrix of rank $r$, and ${\bf n}={\bf n_+}$ be the standard maximal nilpotent subalgebra in the Kac-Moody algebra associated with $A$ (thus, ${\bf n}$ is generated by $E_1,\ldots,E_r$ subject to the Serre relations). Let $\hat U_q({\bf n})$ be the completion (with respect to the natural grading) of the quantized enveloping algebra of ${\bf n}$. For a sequence ${\bf i}=(i_1,\ldots,i_m)$ with $1\le i_k\le r$, let $P_{\bf i}$ be a skew polynomial algebra generated by $t_1,\ldots,t_m$ subject to the relations $t_lt_k=q^{C_{i_k,i_l}}t_kt_l$ ($1\le k<l\le m$) where $C=(C_{ij})=(d_ia_{ij})$ is the symmetric matrix corresponding to $A$. We construct a group-like element ${\bf e}_{\bf i}\in P_{\bf i}\bigotimes \hat U_q({\bf n})$. This element gives rise to the evaluation homomorphism $ψ_{\bf i}:{\bf C}_q[N]\to P_{\bf i}$ given by $ψ_{\bf i}(x)=x({\bf e}_{\bf i})$, where ${\bf C}_q[N]=U_q({\bf n})^0$ is the restricted dual of $U_q({\bf n})$. Under a well-known isomorphism of algebras ${\bf C}_q[N]$ and $U_q({\bf n})$, the map $ψ_{\bf i}$ identifies with Feigin's homomorphism $Φ({\bf i}): U_q({\bf n})\to P_{\bf i}$. We prove that the image of $ψ_{\bf i}$ generates the skew-field of fractions ${\cal F}(P_{\bf i})$ if and only if ${\bf i}$ is a reduced expression of some element $w$ in the Weyl group $W$; furthermore, in the latter case, ${\rm Ker}~ψ_{\bf i}$ depends only on $w$ (so we denote $I_w:={\rm Ker}~ψ_{\bf i}$). This result generalizes the results in [5], [6] to the case of Kac-Moody algebras. We also construct an element ${\cal R}_w\in \big({\bf C}_q[N]/I_w\big)\bigotimes \hat U_q({\bf n})$ which specializes to ${\bf e}_{\bf i}$ under the embedding ${\bf C}_q[N]/I_w\hookrightarrow P_{\bf i}$. The elements ${\cal R}_w$ are closely