Quantum Dynamical coBoundary Equation for finite dimensional simple Lie algebras

For a finite dimensional simple Lie algebra gg, the standard universal solution R(x)∈Uq(g)⊗2R(x)∈Uq(g)⊗2 of the Quantum Dynamical Yang–Baxter Equation quantizes the standard trigonometric solution of the Classical Dynamical Yang–Baxter Equation. It can be built from the standard R  -matrix and from the solution F(x)∈Uq(g)⊗2F(x)∈Uq(g)⊗2 of the Quantum Dynamical coCycle Equation as R(x)=F21−1(x)RF12(x). F(x)F(x) can be computed explicitly as an infinite product through the use of an auxiliary linear equation, the ABRR equation.Inspired by explicit results in the fundamental representation, it has been conjectured that, in the case where g=sl(n+1)g=sl(n+1)(n⩾1)(n⩾1) only, there could exist an element M(x)∈Uq(sl(n+1))M(x)∈Uq(sl(n+1)) such that the dynamical gauge transform RJRJ of R(x)R(x) by M(x)M(x),RJ=M1(x)−1M2(xqh1)−1R(x)M1(xqh2)M2(x),RJ=M1(x)−1M2(xqh1)−1R(x)M1(xqh2)M2(x), does not depend on x   and is a universal solution of the Quantum Yang–Baxter Equation. In the fundamental representation, RJRJ corresponds to the standard solution R   for n=1n=1 and to Cremmer–Gervais's one R12J=J21−1R12J12 for n>1n>1. For consistency, M(x)M(x) should therefore satisfy the Quantum Dynamical coBoundary Equation, i.e.F(x)=Δ(M(x))JM2(x)−1(M1(xqh2))−1,F(x)=Δ(M(x))JM2(x)−1(M1(xqh2))−1, in which J∈Uq(sl(n+1))⊗2J∈Uq(sl(n+1))⊗2 is the universal cocycle associated to 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)M(x), we construct M(x)M(x) in Uq(sl(n+1))Uq(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.