Quantum Dynamical coBoundary Equation for finite dimensional simple Lie algebras

Abstract

For a finite dimensional simple Lie algebra g, the standard universal solution R(x) in $U_q(g)^{\otimes 2}$ of the Quantum Dynamical Yang--Baxter Equation can be built from the standard R--matrix and from the solution F(x) in $U_q(g)^{\otimes 2}$ of the Quantum Dynamical coCycle Equation as $R(x)=F^{-1}_{21}(x) R F_{12}(x).$ It has been conjectured that, in the case where g=sl(n+1) n greater than 1 only, there could exist an element M(x) in $U_q(sl(n+1))$ such that $F(x)=Δ(M(x)){J} M_2(x)^{-1}(M_1(xq^{h_2}))^{-1},$ in which $J\in U_q(sl(n+1))^{\otimes 2}$ is the universal cocycle associated to the Cremmer--Gervais's solution. The aim of this article is to prove this conjecture and to study the properties of the solutions of the Quantum Dynamical coBoundary Equation. In particular, by introducing new basic algebraic objects which are the building blocks of the Gauss decomposition of M(x), we construct M(x) in $U_q(sl(n+1))$ as an explicit infinite product which converges in every finite dimensional representation. We emphasize the relations between these basic objects and some Non Standard Loop algebras and exhibit relations with the dynamical quantum Weyl group.